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Reading & WritingLesson13 minMedium

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Match the relationship between ideas before looking at transition choices.

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YOUR NOTES PROGRESS0 of 496 lessons

Translate between radicals and fractional exponents and simplify perfect-power factors.

Read notes

a⁰ = 1

Explain this formula +

SYMBOL GUIDE

a ≠ 0

WHEN TO USE IT

Use whenever a nonzero base has exponent 0.

WORKED MICRO-EXAMPLE

(−7)⁰.

COMMON SAT TRAP

The result is not 0.

Advanced MathKNOW THIS

Exponents

Negative exponent

a⁻ⁿ = 1/aⁿ

Explain this formula +

SYMBOL GUIDE

a ≠ 0

WHEN TO USE IT

Use to rewrite with positive exponents.

WORKED MICRO-EXAMPLE

x⁻³.

1/x³

COMMON SAT TRAP

A negative exponent creates a reciprocal; it does not make the value negative.

Advanced MathKNOW THIS

Radicals and exponents

i² = −1

Explain this formula +

SYMBOL GUIDE

i = √(−1)

WHEN TO USE IT

Use to simplify square roots of negative numbers and powers of i.

WORKED MICRO-EXAMPLE

√(−49) = 7i.

COMMON SAT TRAP

√(−49) is not −7.

Advanced MathKNOW THIS

Complex numbers

Cycle of powers of i

i, −1, −i, 1 repeat every 4 powers

Explain this formula +

SYMBOL GUIDE

Reduce the exponent modulo 4

WHEN TO USE IT

Use to simplify a large power of i.

WORKED MICRO-EXAMPLE

i²³ has remainder 3 when 23 is divided by 4.

i²³ = −i

COMMON SAT TRAP

A remainder of 0 corresponds to 1.

Problem-Solving & Data AnalysisKNOW THIS

Percents

Percent of a quantity

Part = (percent/100) × whole

Explain this formula +

SYMBOL GUIDE

Part and whole use the same units

WHEN TO USE IT

Use to calculate a stated percentage of an amount.

WORKED MICRO-EXAMPLE

Find 18% of 250.

0.18(250) = 45

COMMON SAT TRAP

Convert the percent to a decimal.

Problem-Solving & Data AnalysisKNOW THIS

Percents

Find the percent

Percent = (part/whole) × 100%

Explain this formula +

SYMBOL GUIDE

Whole = reference amount

WHEN TO USE IT

Use when comparing a part with its original whole.

WORKED MICRO-EXAMPLE

45 is what percent of 250?

18%

COMMON SAT TRAP

Choose the correct reference whole.

Problem-Solving & Data AnalysisKNOW THIS

Percents

Percent change

(new − original)/original × 100%

Explain this formula +

SYMBOL GUIDE

Original value is the denominator

WHEN TO USE IT

Use for relative increase or decrease.

WORKED MICRO-EXAMPLE

A price rises from 80 to 92.

15% increase

COMMON SAT TRAP

Divide by the original, not the new value.

Problem-Solving & Data AnalysisKNOW THIS

Percents

Reverse percentage

Original = final/(1 ± r)

Explain this formula +

SYMBOL GUIDE

Use + for growth and − for decay

WHEN TO USE IT

Use when the post-change value is known.

WORKED MICRO-EXAMPLE

After a 20% discount, a price is $64.

Original = 64/0.8 = $80

COMMON SAT TRAP

Do not subtract 20% of the final value.

Problem-Solving & Data AnalysisKNOW THIS

Percents

Percentage-point change

New percent − old percent

Explain this formula +

SYMBOL GUIDE

Percentage points compare two percentages directly

WHEN TO USE IT

Use when a rate moves from one percent to another.

WORKED MICRO-EXAMPLE

A rate rises from 30% to 42%.

12 percentage points

COMMON SAT TRAP

The relative percent increase would be 40%; that is a different question.

Problem-Solving & Data AnalysisKNOW THIS

Ratios and proportions

Proportion cross-product

a/b = c/d ⇔ ad = bc

Explain this formula +

SYMBOL GUIDE

b and d are nonzero

WHEN TO USE IT

Use to solve equivalent-ratio relationships.

WORKED MICRO-EXAMPLE

3/5 = x/40.

5x = 120, so x = 24

COMMON SAT TRAP

Match corresponding quantities before cross-multiplying.

Problem-Solving & Data AnalysisKNOW THIS

Rates

Unit rate

Unit rate = quantity/1 unit

Explain this formula +

SYMBOL GUIDE

A rate compares quantities with different units

WHEN TO USE IT

Use to compare prices, speed, density, and productivity.

WORKED MICRO-EXAMPLE

$18 for 6 notebooks.

$3 per notebook

COMMON SAT TRAP

Keep the unit order requested by the question.

Problem-Solving & Data AnalysisKNOW THIS

Rates

Distance-rate-time

d = rt

Explain this formula +

SYMBOL GUIDE

d = distance; r = rate; t = time

WHEN TO USE IT

Use for constant-speed travel.

WORKED MICRO-EXAMPLE

At 55 mph for 3 hours.

d = 165 miles

COMMON SAT TRAP

Make time and rate units compatible.

Problem-Solving & Data AnalysisKNOW THIS

Rates

Density

density = mass/volume

Explain this formula +

SYMBOL GUIDE

Mass and volume units determine density units

WHEN TO USE IT

Use when mass is distributed through a volume.

WORKED MICRO-EXAMPLE

A 240 g object occupies 80 cm³.

3 g/cm³

COMMON SAT TRAP

Do not invert mass and volume.

Problem-Solving & Data AnalysisKNOW THIS

Statistics

Weighted mean

Weighted mean = Σ(value × weight)/Σweights

Explain this formula +

SYMBOL GUIDE

Weights may be counts, credits, or probabilities

WHEN TO USE IT

Use when observations contribute unequally.

WORKED MICRO-EXAMPLE

Scores 80 and 95 have weights 2 and 3.

(80·2 + 95·3)/5 = 89

COMMON SAT TRAP

Divide by total weight, not number of categories.

Problem-Solving & Data AnalysisKNOW THIS

Statistics

Arithmetic mean

mean = sum of values/number of values

Explain this formula +

SYMBOL GUIDE

The mean is the balance point

WHEN TO USE IT

Use to calculate or update an average.

WORKED MICRO-EXAMPLE

For 4, 7, 9, 10, sum = 30 and n = 4.

Mean = 7.5

COMMON SAT TRAP

Count every data value, including repeats.

Problem-Solving & Data AnalysisKNOW THIS

Statistics

Recover a missing value from a mean

Missing value = mean × count − known sum

Explain this formula +

SYMBOL GUIDE

Mean × count gives total sum

WHEN TO USE IT

Use when an average and all but one value are given.

WORKED MICRO-EXAMPLE

Five values average 12; four sum to 49.

Missing value = 60 − 49 = 11

COMMON SAT TRAP

Multiply the mean by the total number of values first.

Problem-Solving & Data AnalysisKNOW THIS

Statistics

Range

Range = maximum − minimum

Explain this formula +

SYMBOL GUIDE

Range measures total spread

WHEN TO USE IT

Use for the simplest measure of variability.

WORKED MICRO-EXAMPLE

For 3, 8, 8, 14.

Range = 11

COMMON SAT TRAP

Range is not the number of distinct values.

Problem-Solving & Data AnalysisKNOW THIS

Statistics

Interquartile range

IQR = Q₃ − Q₁

Explain this formula +

SYMBOL GUIDE

Q₁ and Q₃ bound the middle 50%

WHEN TO USE IT

Use to compare spread while reducing outlier influence.

WORKED MICRO-EXAMPLE

If Q₁ = 18 and Q₃ = 31.

IQR = 13

COMMON SAT TRAP

Do not subtract minimum from maximum.

Problem-Solving & Data AnalysisKNOW THIS

Probability

Basic probability

P(event) = favorable outcomes/total outcomes

Explain this formula +

SYMBOL GUIDE

Outcomes are equally likely in the basic model

WHEN TO USE IT

Use to calculate the chance of a specified event.

WORKED MICRO-EXAMPLE

Three of 12 equal sectors are blue.

P(blue) = 3/12 = 1/4

COMMON SAT TRAP

The denominator contains all possible outcomes.

Problem-Solving & Data AnalysisKNOW THIS

Probability

Complement rule

P(not A) = 1 − P(A)

Explain this formula +

SYMBOL GUIDE

A and not A cover every outcome

WHEN TO USE IT

Use when the opposite event is easier.

WORKED MICRO-EXAMPLE

If P(rain)=0.35.

P(no rain)=0.65

COMMON SAT TRAP

Probabilities must use the same scale: decimals with decimals.

Problem-Solving & Data AnalysisKNOW THIS

Probability

General addition rule

P(A or B) = P(A) + P(B) − P(A and B)

Explain this formula +

SYMBOL GUIDE

Subtract overlap counted twice

WHEN TO USE IT

Use for a union of possibly overlapping events.

WORKED MICRO-EXAMPLE

P(A)=0.5, P(B)=0.4, P(A and B)=0.2.

P(A or B)=0.7

COMMON SAT TRAP

Add directly only when events are mutually exclusive.

Problem-Solving & Data AnalysisKNOW THIS

Probability

Independent-event multiplication

P(A and B) = P(A)P(B)

Explain this formula +

SYMBOL GUIDE

One event does not change the other

WHEN TO USE IT

Use for the intersection of independent events.

WORKED MICRO-EXAMPLE

P(head)=1/2 on each of two tosses.

P(two heads)=1/4

COMMON SAT TRAP

Do not multiply if the first event changes the second probability.

Problem-Solving & Data AnalysisKNOW THIS

Probability

Conditional probability

P(A|B) = P(A and B)/P(B)

Explain this formula +

SYMBOL GUIDE

The sample space is restricted to B

WHEN TO USE IT

Use when the question says “given that.”

WORKED MICRO-EXAMPLE

20 students play music; 8 of them also play sports.

P(sports | music)=8/20=0.4

COMMON SAT TRAP

The denominator is the given group.

Problem-Solving & Data AnalysisKNOW THIS

Probability

Expected value

E(X) = Σ(value × probability)

Explain this formula +

SYMBOL GUIDE

Multiply each possible value by its probability

WHEN TO USE IT

Use for long-run average payoff or outcome.

WORKED MICRO-EXAMPLE

Win $10 with probability 0.2 and $0 otherwise.

E = $2

COMMON SAT TRAP

Expected value need not be a possible single outcome.

Problem-Solving & Data AnalysisKNOW THIS

Two-way tables

Relative frequency

Relative frequency = category count/total count

Explain this formula +

SYMBOL GUIDE

Can be joint, marginal, or conditional

WHEN TO USE IT

Use to turn table counts into proportions.

WORKED MICRO-EXAMPLE

18 of 60 respondents choose A.

Relative frequency = 0.30

COMMON SAT TRAP

For a conditional frequency, use the relevant row or column total.

Problem-Solving & Data AnalysisKNOW THIS

Sampling

Sample proportion

p̂ = successes/sample size

Explain this formula +

SYMBOL GUIDE

p̂ estimates a population proportion

WHEN TO USE IT

Use to summarize binary survey or experiment data.

WORKED MICRO-EXAMPLE

84 of 120 sampled voters support a measure.

p̂ = 0.70

COMMON SAT TRAP

A sample statistic is an estimate, not proof of the exact population value.

Problem-Solving & Data AnalysisKNOW THIS

Data modeling

Residual

Residual = observed y − predicted ŷ

Explain this formula +

SYMBOL GUIDE

Positive residual means point lies above model

WHEN TO USE IT

Use to measure a model’s prediction error.

WORKED MICRO-EXAMPLE

Observed 52, predicted 47.

Residual = 5

COMMON SAT TRAP

Subtract predicted from observed in that order.

Problem-Solving & Data AnalysisKNOW THIS

Data modeling

Linear model prediction

ŷ = mx + b

Explain this formula +

SYMBOL GUIDE

ŷ = predicted response

WHEN TO USE IT

Use a line of best fit to estimate an outcome.

WORKED MICRO-EXAMPLE

For ŷ = 3.2x + 11 and x = 5.

ŷ = 27

COMMON SAT TRAP

A prediction outside the data range is extrapolation and may be unreliable.

Problem-Solving & Data AnalysisKNOW THIS

Financial models

Simple interest

I = Prt and A = P(1 + rt)

Explain this formula +

SYMBOL GUIDE

P = principal; r = annual decimal rate; t = years

WHEN TO USE IT

Use when interest is calculated only on the original principal.

WORKED MICRO-EXAMPLE

$500 at 4% simple interest for 3 years.

I = $60; A = $560

COMMON SAT TRAP

Simple interest does not compound.

Problem-Solving & Data AnalysisKNOW THIS

Mixtures

Mixture amount balance

amount₁ + amount₂ = total amount

Explain this formula +

SYMBOL GUIDE

Concentration × amount gives pure component

WHEN TO USE IT

Use with a second equation for mixture problems.

WORKED MICRO-EXAMPLE

x liters of 20% solution and y liters of 50% solution make 10 liters.

x + y = 10

COMMON SAT TRAP

The concentration equation is 0.20x + 0.50y = pure amount, not total volume.

Problem-Solving & Data AnalysisKNOW THIS

Units

Unit conversion

new quantity = old quantity × conversion factor

Explain this formula +

SYMBOL GUIDE

Write the factor so unwanted units cancel

WHEN TO USE IT

Use for length, time, currency, and compound units.

WORKED MICRO-EXAMPLE

72 miles/hour × 1 hour/60 minutes.

1.2 miles/minute

COMMON SAT TRAP

Arrange the conversion fraction by unit cancellation, not memory.

Problem-Solving & Data AnalysisKNOW THIS

Scale drawings

Length scale factor

new length = k × original length

Explain this formula +

SYMBOL GUIDE

k = linear scale factor

WHEN TO USE IT

Use for similar figures and maps.

WORKED MICRO-EXAMPLE

A drawing uses scale 1:50; 3 cm represents.

150 cm

COMMON SAT TRAP

A 1:50 scale means multiply drawing length by 50 for actual length.

Problem-Solving & Data AnalysisKNOW THIS

Scale drawings

Area scale factor

new area = k² × original area

Explain this formula +

SYMBOL GUIDE

k = linear scale factor

WHEN TO USE IT

Use when similar figures are enlarged or reduced.

WORKED MICRO-EXAMPLE

Side lengths double.

Area becomes 4 times as large

COMMON SAT TRAP

Area does not scale by k.

Problem-Solving & Data AnalysisKNOW THIS

Scale drawings

Volume scale factor

new volume = k³ × original volume

Explain this formula +

SYMBOL GUIDE

k = linear scale factor

WHEN TO USE IT

Use for similar three-dimensional solids.

WORKED MICRO-EXAMPLE

All dimensions triple.

Volume becomes 27 times as large

COMMON SAT TRAP

Volume scales by the cube of the length factor.

Geometry & TrigonometryBLUEBOOK PROVIDES

Area

Area of a rectangle

A = lw

Explain this formula +

SYMBOL GUIDE

l = length; w = width

WHEN TO USE IT

Use for rectangles and as a base area inside other solids.

WORKED MICRO-EXAMPLE

A rectangle is 12 by 7.

A = 84

COMMON SAT TRAP

Area uses square units.

Geometry & TrigonometryKNOW THIS

Perimeter

Perimeter of a rectangle

P = 2l + 2w

Explain this formula +

SYMBOL GUIDE

l = length; w = width

WHEN TO USE IT

Use for total boundary length.

WORKED MICRO-EXAMPLE

A rectangle is 12 by 7.

P = 38

COMMON SAT TRAP

Perimeter and area answer different questions.

Geometry & TrigonometryKNOW THIS

Area

Area of a square

A = s²

Explain this formula +

SYMBOL GUIDE

s = side length

WHEN TO USE IT

Use when all four sides are equal and angles are right angles.

WORKED MICRO-EXAMPLE

A square has side 9.

A = 81

COMMON SAT TRAP

Do not multiply by 4; that gives perimeter.

Geometry & TrigonometryKNOW THIS

Perimeter

Perimeter of a square

P = 4s

Explain this formula +

SYMBOL GUIDE

s = side length

WHEN TO USE IT

Use for the boundary of a square.

WORKED MICRO-EXAMPLE

A square has side 9.

P = 36

COMMON SAT TRAP

Keep linear units for perimeter.

Geometry & TrigonometryKNOW THIS

Area

Area of a parallelogram

A = bh

Explain this formula +

SYMBOL GUIDE

b = base; h = perpendicular height

WHEN TO USE IT

Use for any parallelogram.

WORKED MICRO-EXAMPLE

Base 10 and perpendicular height 6.

A = 60

COMMON SAT TRAP

A slanted side is not the height unless perpendicular to the base.

Geometry & TrigonometryBLUEBOOK PROVIDES

Area

Area of a triangle

A = ½bh

Explain this formula +

SYMBOL GUIDE

b = base; h = perpendicular height

WHEN TO USE IT

Use for any triangle when a base-height pair is known.

WORKED MICRO-EXAMPLE

Base 14 and height 9.

A = 63

COMMON SAT TRAP

The height must be perpendicular to the chosen base.

Geometry & TrigonometryKNOW THIS

Area

Area of a trapezoid

A = ½(b₁ + b₂)h

Explain this formula +

SYMBOL GUIDE

b₁ and b₂ = parallel bases; h = perpendicular height

WHEN TO USE IT

Use for a quadrilateral with one pair of parallel sides.

WORKED MICRO-EXAMPLE

Bases 8 and 14, height 5.

A = 55

COMMON SAT TRAP

Add the parallel bases before multiplying by one-half the height.

Geometry & TrigonometryBLUEBOOK PROVIDES

Angles

Triangle angle sum

A + B + C = 180°

Explain this formula +

SYMBOL GUIDE

A, B, C = interior angles

WHEN TO USE IT

Use to find a missing triangle angle.

WORKED MICRO-EXAMPLE

Angles are 42° and 68°.

Third angle = 70°

COMMON SAT TRAP

Use interior angles, not an exterior angle.

Geometry & TrigonometryKNOW THIS

Polygons

Interior-angle sum of a polygon

Sum = (n − 2)180°

Explain this formula +

SYMBOL GUIDE

n = number of sides

WHEN TO USE IT

Use to total all interior angles of a polygon.

WORKED MICRO-EXAMPLE

For a hexagon, n = 6.

Sum = 720°

COMMON SAT TRAP

Subtract 2 before multiplying.

Geometry & TrigonometryKNOW THIS

Polygons

One interior angle of a regular polygon

Angle = (n − 2)180°/n

Explain this formula +

SYMBOL GUIDE

n = number of equal sides/angles

WHEN TO USE IT

Use only when the polygon is regular.

WORKED MICRO-EXAMPLE

A regular octagon.

Angle = 135°

COMMON SAT TRAP

Divide by n only because every interior angle is equal.

Geometry & TrigonometryKNOW THIS

Polygons

Exterior-angle sum

Sum of one exterior angle at each vertex = 360°

Explain this formula +

SYMBOL GUIDE

Works for any convex polygon

WHEN TO USE IT

Use to total a full turn around a polygon.

WORKED MICRO-EXAMPLE

Five exterior angles include 70°, 80°, 65°, 75°, and x.

x = 70°

COMMON SAT TRAP

Use one exterior angle per vertex.

Geometry & TrigonometryKNOW THIS

Polygons

One exterior angle of a regular polygon

Exterior angle = 360°/n

Explain this formula +

SYMBOL GUIDE

n = number of sides

WHEN TO USE IT

Use to connect a regular polygon’s side count and turning angle.

WORKED MICRO-EXAMPLE

A regular 12-gon.

Exterior angle = 30°

COMMON SAT TRAP

Interior and exterior angles form a linear pair.

Geometry & TrigonometryBLUEBOOK PROVIDES

Circles

Area of a circle

A = πr²

Explain this formula +

SYMBOL GUIDE

r = radius

WHEN TO USE IT

Use for the region inside a circle.

WORKED MICRO-EXAMPLE

Radius 6.

A = 36π

COMMON SAT TRAP

Square the radius, not π.

Geometry & TrigonometryBLUEBOOK PROVIDES

Circles

Circumference of a circle

C = 2πr = πd

Explain this formula +

SYMBOL GUIDE

r = radius; d = diameter

WHEN TO USE IT

Use for distance around a circle.

WORKED MICRO-EXAMPLE

Diameter 10.

C = 10π

COMMON SAT TRAP

If diameter is given, do not double it again.

Geometry & TrigonometryKNOW THIS

Circles

Diameter-radius relationship

d = 2r

Explain this formula +

SYMBOL GUIDE

d = diameter; r = radius

WHEN TO USE IT

Use to switch between the two circle measurements.

WORKED MICRO-EXAMPLE

Diameter 18.

Radius 9

COMMON SAT TRAP

The diameter crosses the full circle through the center.

Geometry & TrigonometryBLUEBOOK PROVIDES

Circles

Degrees in a circle

One full circle = 360°

Explain this formula +

SYMBOL GUIDE

A full rotation measures 360 degrees

WHEN TO USE IT

Use for arcs, central angles, sectors, and rotations.

WORKED MICRO-EXAMPLE

A quarter circle has central angle 360°/4.

90°

COMMON SAT TRAP

A semicircle is 180°, not 360°.

Geometry & TrigonometryBLUEBOOK PROVIDES

Circles

Radians in a circle

One full circle = 2π radians

Explain this formula +

SYMBOL GUIDE

π radians = 180°

WHEN TO USE IT

Use to connect radian and degree measures.

WORKED MICRO-EXAMPLE

A half-circle is half of 2π.

π radians

COMMON SAT TRAP

Radians are angle measures; do not attach degree symbols.

Geometry & TrigonometryKNOW THIS

Angles

Degree-radian conversion

radians = degrees × π/180

Explain this formula +

SYMBOL GUIDE

π radians = 180°

WHEN TO USE IT

Use to convert a degree angle to radians.

WORKED MICRO-EXAMPLE

Convert 60°.

π/3 radians

COMMON SAT TRAP

Cancel the degree factor before simplifying.

Geometry & TrigonometryKNOW THIS

Angles

Radian-degree conversion

degrees = radians × 180/π

Explain this formula +

SYMBOL GUIDE

π radians = 180°

WHEN TO USE IT

Use to convert radians to degrees.

WORKED MICRO-EXAMPLE

Convert 5π/6.

150°

COMMON SAT TRAP

The π often cancels.

Geometry & TrigonometryKNOW THIS

Circles

Arc length with degrees

s = (θ/360°)(2πr)

Explain this formula +

SYMBOL GUIDE

θ = central angle; r = radius

WHEN TO USE IT

Use when a central angle in degrees cuts off an arc.

WORKED MICRO-EXAMPLE

Radius 9, central angle 80°.

s = 4π

COMMON SAT TRAP

Arc length is a fraction of circumference, not area.

Geometry & TrigonometryKNOW THIS

Circles

Arc length with radians

s = rθ

Explain this formula +

SYMBOL GUIDE

θ must be in radians

WHEN TO USE IT

Use the fast arc formula when the angle is already in radians.

WORKED MICRO-EXAMPLE

r = 6 and θ = 2π/3.

s = 4π

COMMON SAT TRAP

This form does not work with degree input.

Geometry & TrigonometryKNOW THIS

Circles

Sector area with degrees

A = (θ/360°)(πr²)

Explain this formula +

SYMBOL GUIDE

θ = central angle

WHEN TO USE IT

Use for a slice of a circle when θ is in degrees.

WORKED MICRO-EXAMPLE

Radius 12, central angle 30°.

A = 12π

COMMON SAT TRAP

Sector area is a fraction of circle area.

Geometry & TrigonometryKNOW THIS

Circles

Sector area with radians

A = ½r²θ

Explain this formula +

SYMBOL GUIDE

θ must be in radians

WHEN TO USE IT

Use for sector area with a radian angle.

WORKED MICRO-EXAMPLE

r = 4 and θ = π/2.

A = 4π

COMMON SAT TRAP

Do not use this formula with degrees.

Geometry & TrigonometryKNOW THIS

Coordinate circles

Equation of a circle

(x − h)² + (y − k)² = r²

Explain this formula +

SYMBOL GUIDE

(h, k) = center; r = radius

WHEN TO USE IT

Use to read or build a circle in the coordinate plane.

WORKED MICRO-EXAMPLE

(x + 2)² + (y − 5)² = 49.

Center (−2, 5), radius 7

COMMON SAT TRAP

The center signs are opposite the signs inside the equation.

Geometry & TrigonometryKNOW THIS

Circles

Central angle and intercepted arc

central angle measure = intercepted arc measure

Explain this formula +

SYMBOL GUIDE

Vertex of central angle is the center

WHEN TO USE IT

Use to move directly between a central angle and its minor arc.

WORKED MICRO-EXAMPLE

A central angle is 110°.

Its intercepted minor arc is 110°

COMMON SAT TRAP

This one-to-one rule is not the inscribed-angle rule.

Geometry & TrigonometryKNOW THIS

Circles

Inscribed angle theorem

inscribed angle = ½(intercepted arc)

Explain this formula +

SYMBOL GUIDE

Vertex lies on the circle

WHEN TO USE IT

Use when an angle on the circle intercepts an arc.

WORKED MICRO-EXAMPLE

An inscribed angle intercepts a 140° arc.

Angle = 70°

COMMON SAT TRAP

Identify the intercepted arc opposite the angle.

Geometry & TrigonometryKNOW THIS

Circles

Radius-tangent angle

radius ⟂ tangent at point of tangency

Explain this formula +

SYMBOL GUIDE

The angle is 90°

WHEN TO USE IT

Use when a tangent touches a circle at one point.

WORKED MICRO-EXAMPLE

OT is a radius and line l is tangent at T.

OT forms a 90° angle with l

COMMON SAT TRAP

The perpendicular relationship occurs at the tangency point.

Geometry & TrigonometryKNOW THIS

Circles

Angle in a semicircle

An inscribed angle subtending a diameter = 90°

Explain this formula +

SYMBOL GUIDE

The intercepted arc is 180°

WHEN TO USE IT

Use when an inscribed triangle has a diameter as one side.

WORKED MICRO-EXAMPLE

AB is a diameter and C lies on the circle.

∠ACB = 90°

COMMON SAT TRAP

The right angle is at the point on the circle, not necessarily at the center.

Geometry & TrigonometryKNOW THIS

Circles

Intersecting chords theorem

AP·PB = CP·PD

Explain this formula +

SYMBOL GUIDE

Two chords intersect inside a circle

WHEN TO USE IT

Use to solve segment lengths formed by intersecting chords.

WORKED MICRO-EXAMPLE

One chord has segments 3 and 8; the other 4 and x.

24 = 4x, so x = 6

COMMON SAT TRAP

Multiply the two pieces of the same chord.

Geometry & TrigonometryBLUEBOOK PROVIDES

Right triangles

Pythagorean theorem

c² = a² + b²

Explain this formula +

SYMBOL GUIDE

c = hypotenuse; a and b = legs

WHEN TO USE IT

Use only in right triangles.

WORKED MICRO-EXAMPLE

Legs 6 and 8.

c = 10

COMMON SAT TRAP

The hypotenuse is opposite the right angle and is always c.

Geometry & TrigonometryBLUEBOOK PROVIDES

Special right triangles

45°-45°-90° triangle

legs s, s; hypotenuse s√2

Explain this formula +

SYMBOL GUIDE

The two legs are equal

WHEN TO USE IT

Use for an isosceles right triangle.

WORKED MICRO-EXAMPLE

A leg is 7.

Hypotenuse = 7√2

COMMON SAT TRAP

The hypotenuse is the side multiplied by √2.

Geometry & TrigonometryBLUEBOOK PROVIDES

Special right triangles

30°-60°-90° triangle

short leg x; long leg x√3; hypotenuse 2x

Explain this formula +

SYMBOL GUIDE

Short leg is opposite 30°

WHEN TO USE IT

Use for a right triangle containing 30° and 60°.

WORKED MICRO-EXAMPLE

Short leg is 5.

Long leg = 5√3; hypotenuse = 10

COMMON SAT TRAP

Label sides by the angles opposite them before using the ratio.

Geometry & TrigonometryKNOW THIS

Right-triangle trigonometry

Sine ratio

sin θ = opposite/hypotenuse

Explain this formula +

SYMBOL GUIDE

θ = acute reference angle

WHEN TO USE IT

Use when opposite and hypotenuse are involved.

WORKED MICRO-EXAMPLE

Opposite = 6 and hypotenuse = 10.

sin θ = 3/5

COMMON SAT TRAP

Opposite and adjacent depend on the chosen angle.

Geometry & TrigonometryKNOW THIS

Right-triangle trigonometry

Cosine ratio

cos θ = adjacent/hypotenuse

Explain this formula +

SYMBOL GUIDE

θ = acute reference angle

WHEN TO USE IT

Use when adjacent and hypotenuse are involved.

WORKED MICRO-EXAMPLE

Adjacent = 8 and hypotenuse = 10.

cos θ = 4/5

COMMON SAT TRAP

The hypotenuse never changes with the reference angle.

Geometry & TrigonometryKNOW THIS

Right-triangle trigonometry

Tangent ratio

tan θ = opposite/adjacent

Explain this formula +

SYMBOL GUIDE

θ = acute reference angle

WHEN TO USE IT

Use when the two legs are involved.

WORKED MICRO-EXAMPLE

Opposite = 6 and adjacent = 8.

tan θ = 3/4

COMMON SAT TRAP

Tangent does not use the hypotenuse.

Geometry & TrigonometryKNOW THIS

Trigonometry

Pythagorean trig identity

sin²θ + cos²θ = 1

Explain this formula +

SYMBOL GUIDE

Valid for every angle where sine and cosine are defined

WHEN TO USE IT

Use to find sine from cosine or vice versa.

WORKED MICRO-EXAMPLE

If cos θ = 12/13 and θ is acute.

sin θ = 5/13

COMMON SAT TRAP

Take the positive root only when the angle is known to be acute.

Geometry & TrigonometryKNOW THIS

Similarity

Similar-figure side ratio

corresponding side ratios are equal

Explain this formula +

SYMBOL GUIDE

Matching angles identify matching sides

WHEN TO USE IT

Use to find missing lengths in similar figures.

WORKED MICRO-EXAMPLE

A side 6 corresponds to 9; side 10 corresponds to x.

6/9 = 10/x, so x = 15

COMMON SAT TRAP

Write every ratio in the same figure order.

Geometry & TrigonometryKNOW THIS

Similarity

Side-splitter theorem

parallel line creates proportional side segments

Explain this formula +

SYMBOL GUIDE

A segment parallel to one triangle side creates a smaller similar triangle

WHEN TO USE IT

Use when a line inside a triangle is parallel to a side.

WORKED MICRO-EXAMPLE

Scale from small to large is 2:5; a small side is 8.

Corresponding large side = 20

COMMON SAT TRAP

Confirm the interior segment is marked parallel.

Geometry & TrigonometryKNOW THIS

Volume

Volume of any prism

V = Bh

Explain this formula +

SYMBOL GUIDE

B = base area; h = perpendicular length/height

WHEN TO USE IT

Use for solids with congruent parallel bases.

WORKED MICRO-EXAMPLE

Base area 18 and prism length 7.

V = 126

COMMON SAT TRAP

B is an area, not a base edge length.

Geometry & TrigonometryBLUEBOOK PROVIDES

Volume

Volume of a rectangular prism

V = lwh

Explain this formula +

SYMBOL GUIDE

l, w, h = perpendicular dimensions

WHEN TO USE IT

Use for boxes and cuboids.

WORKED MICRO-EXAMPLE

Dimensions 3, 5, and 8.

V = 120

COMMON SAT TRAP

Volume uses cubic units.

Geometry & TrigonometryKNOW THIS

Volume

Volume of a cube

V = s³

Explain this formula +

SYMBOL GUIDE

s = edge length

WHEN TO USE IT

Use when all three dimensions are equal.

WORKED MICRO-EXAMPLE

Edge 4.

V = 64

COMMON SAT TRAP

Cubing is multiplying the side three times.

Geometry & TrigonometryBLUEBOOK PROVIDES

Volume

Volume of a cylinder

V = πr²h

Explain this formula +

SYMBOL GUIDE

r = base radius; h = perpendicular height

WHEN TO USE IT

Use for a circular prism.

WORKED MICRO-EXAMPLE

r = 3 and h = 10.

V = 90π

COMMON SAT TRAP

Square the radius before multiplying by height.

Geometry & TrigonometryBLUEBOOK PROVIDES

Volume

Volume of a pyramid

V = ⅓Bh

Explain this formula +

SYMBOL GUIDE

B = base area; h = perpendicular height

WHEN TO USE IT

Use for any pyramid.

WORKED MICRO-EXAMPLE

Base area 48 and height 9.

V = 144

COMMON SAT TRAP

Include the factor 1/3.

Geometry & TrigonometryBLUEBOOK PROVIDES

Volume

Volume of a cone

V = ⅓πr²h

Explain this formula +

SYMBOL GUIDE

r = radius; h = perpendicular height

WHEN TO USE IT

Use for a circular pyramid.

WORKED MICRO-EXAMPLE

r = 6 and h = 5.

V = 60π

COMMON SAT TRAP

Use vertical height, not slant height.

Geometry & TrigonometryBLUEBOOK PROVIDES

Volume

Volume of a sphere

V = ⁴⁄₃πr³

Explain this formula +

SYMBOL GUIDE

r = radius

WHEN TO USE IT

Use for the space inside a sphere.

WORKED MICRO-EXAMPLE

r = 3.

V = 36π

COMMON SAT TRAP

Cube the radius and multiply by 4/3.

Geometry & TrigonometryKNOW THIS

Surface area

Surface area of a rectangular prism

SA = 2lw + 2lh + 2wh

Explain this formula +

SYMBOL GUIDE

l, w, h = dimensions

WHEN TO USE IT

Use for the total area of all six faces.

WORKED MICRO-EXAMPLE

l = 4, w = 3, h = 2.

SA = 52

COMMON SAT TRAP

Surface area uses square units, not cubic units.

Geometry & TrigonometryKNOW THIS

Surface area

Surface area of a cube

SA = 6s²

Explain this formula +

SYMBOL GUIDE

s = edge length

WHEN TO USE IT

Use for six congruent square faces.

WORKED MICRO-EXAMPLE

s = 5.

SA = 150

COMMON SAT TRAP

A cube has six faces, not four.

Geometry & TrigonometryKNOW THIS

Surface area

Surface area of a cylinder

SA = 2πr² + 2πrh

Explain this formula +

SYMBOL GUIDE

Two circular bases plus curved rectangle

WHEN TO USE IT

Use for the complete outer surface of a closed cylinder.

WORKED MICRO-EXAMPLE

r = 2 and h = 7.

SA = 36π

COMMON SAT TRAP

Include both circular bases.

Geometry & TrigonometryKNOW THIS

Surface area

Surface area of a sphere

SA = 4πr²

Explain this formula +

SYMBOL GUIDE

r = radius

WHEN TO USE IT

Use for the outside area of a sphere.

WORKED MICRO-EXAMPLE

r = 5.

SA = 100π

COMMON SAT TRAP

Surface area squares the radius; volume cubes it.

Geometry & TrigonometryKNOW THIS

Surface area

Surface area of a cone

SA = πr² + πrl

Explain this formula +

SYMBOL GUIDE

r = radius; l = slant height

WHEN TO USE IT

Use for a closed cone’s base plus curved surface.

WORKED MICRO-EXAMPLE

r = 3 and l = 5.

SA = 24π

COMMON SAT TRAP

Surface area uses slant height; volume uses perpendicular height.

AYLOTI • Sir Hunain Zia • SAT Score 1594/1600 • 154 Total A Grades (Personal), 11 World Records, 7 Distinctions

1,600+ HIGH-UTILITY WORDS

SAT Vocabulary Lab

Learn through context, tone, word parts, precision, testing and spaced review—not a dead alphabetical dump.

COMPLETE WORD BANK4977context-rich words

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WORD OF THE DAYAugust 3

adjective • Foundation

mocking

making fun of someone or something, often with contempt

The essay’s exaggerated imitation creates a mocking tone.

8 PURPOSE-BUILT DECKS

Choose one simple route

Study a deck, review due words, or jump directly into a test. Every route teaches meaning in context.

AYLOTI • Sir Hunain Zia • SAT Score 1594/1600 • 154 Total A Grades (Personal), 11 World Records, 7 Distinctions

100 DECODING TOOLS

full of or having

dubious, zealous, anomalous

suffix

-scope

instrument for viewing

telescope, microscope, endoscope

AYLOTI • Sir Hunain Zia • SAT Score 1594/1600 • 154 Total A Grades (Personal), 11 World Records, 7 Distinctions

48 HIGH-VALUE PAIRS

Words students commonly confuse

Learn the distinction and see both words working in one sentence.

affect / effect

Affect is usually a verb meaning influence.

Effect is usually a noun meaning result.

The policy affected prices; its effect was immediate.

allude / elude

Allude means refer indirectly.

Elude means escape or avoid.

The poem alludes to a myth; its exact meaning eludes some readers.

complement / compliment

Complement means complete or pair well.

Compliment means praise.

The chart complements the text; the editor compliments the clear design.

continual / continuous

Continual means recurring with breaks.

Continuous means without interruption.

Continual alarms occurred during the continuous 12-hour test.

credible / credulous

Credible means believable.

Credulous means too ready to believe.

A credible source should persuade even a non-credulous reader.

discrete / discreet

Discrete means separate.

Discreet means tactful or unobtrusive.

The process has discrete stages; the interview was held discreetly.

emigrate / immigrate

Emigrate means leave a country.

Immigrate means enter a country to live.

They emigrated from Spain and immigrated to Canada.

eminent / imminent

Eminent means distinguished.

Imminent means about to happen.

The eminent scientist warned that the eruption was imminent.

explicit / implicit

Explicit means directly stated.

Implicit means suggested.

The rule is explicit; the criticism is implicit.

farther / further

Farther usually concerns physical distance.

Further usually concerns degree or additional action.

We walked farther to investigate further.

fewer / less

Fewer modifies countable items.

Less modifies uncountable amounts.

Fewer cars produced less pollution.

imply / infer

A speaker or text implies.

A reader or listener infers.

The author implies caution; readers infer that more evidence is needed.

its / it’s

Its is possessive.

It’s means it is or it has.

The device lost its charge; it’s now inactive.

lay / lie

Lay means place something and takes an object.

Lie means recline and takes no object.

Lay the book down, then lie on the couch.

loath / loathe

Loath is an adjective meaning reluctant.

Loathe is a verb meaning hate.

She was loath to admit that she loathed the proposal.

loose / lose

Loose means not tight.

Lose means misplace or fail to keep.

A loose screw may cause the machine to lose power.

principal / principle

Principal means main or a school leader.

Principle means a rule or belief.

The principal reason follows a scientific principle.

rational / rationale

Rational means reasonable.

Rationale means underlying reason.

The choice is rational, and the report explains its rationale.

stationary / stationery

Stationary means not moving.

Stationery means writing materials.

The stationary cart carried boxes of stationery.

than / then

Than makes a comparison.

Then refers to time or consequence.

The first trial was faster than the second; then the team repeated it.

their / there / they’re

Their is possessive.

There indicates place; they’re means they are.

They’re placing their samples over there.

who / whom

Who acts as a subject.

Whom acts as an object.

Who led the team? Whom did the team select?

accept / except

Accept means receive or agree to.

Except means excluding.

All participants except Luis accepted the terms.

adapt / adopt

Adapt means modify for a use.

Adopt means take up or choose.

The school adapted the schedule and adopted a new policy.

adverse / averse

Adverse means harmful or unfavorable.

Averse means opposed or reluctant.

Adverse weather affected hikers who were averse to risk.

advice / advise

Advice is a noun.

Advise is a verb.

The counselor gave advice and advised patience.

cite / site / sight

Cite means quote as evidence.

Site means location; sight relates to seeing.

The report cites a study conducted at the coastal site within sight of the harbor.

conscience / conscious

Conscience is moral awareness.

Conscious means awake or aware.

Her conscience troubled her because she was conscious of the error.

economic / economical

Economic relates to economies or finance.

Economical means inexpensive or efficient.

The economic plan recommends an economical transport system.

elicit / illicit

Elicit means draw out a response.

Illicit means illegal or forbidden.

The question elicited details about the illicit trade.

ensure / insure

Ensure means make certain.

Insure means provide insurance.

Checks ensure accuracy; the policy insures the equipment.

formally / formerly

Formally means officially.

Formerly means previously.

The building formerly housed a bank and was formally reopened as a museum.

historic / historical

Historic means important in history.

Historical means related to the past.

The historic vote is documented in historical records.

ingenious / ingenuous

Ingenious means clever and inventive.

Ingenuous means innocent and sincere.

The ingenious device was explained by its ingenuous young inventor.

precede / proceed

Precede means come before.

Proceed means continue.

A safety check must precede the launch before the team can proceed.

prescribe / proscribe

Prescribe means recommend or order.

Proscribe means prohibit.

Doctors prescribe medicine; regulations proscribe unsafe additives.

prosecute / persecute

Prosecute means bring legal action.

Persecute means mistreat systematically.

The state prosecuted officials accused of persecuting the minority.

respectfully / respectively

Respectfully means with respect.

Respectively links items in the stated order.

Mina and Ali scored 720 and 740, respectively, and both responded respectfully.

sometime / some time / sometimes

Sometime means at an unspecified time.

Some time means a period; sometimes means occasionally.

Sometime this week, spend some time reviewing words you sometimes confuse.

therefore / however

Therefore signals a result.

However signals a contrast.

The sample was biased; therefore, the estimate was unreliable. However, the raw observations remain useful.

which / that

Which often introduces nonessential information.

That usually introduces essential information.

The red book, which is torn, is the one that contains the map.

among / between

Among usually concerns a group.

Between concerns distinct relationships, even among more than two items.

The prize was shared among the team; negotiations occurred between the four departments.

amount / number

Amount modifies uncountable quantities.

Number modifies countable items.

A large amount of rain damaged a number of roads.

assure / ensure

Assure means tell a person confidently.

Ensure means make an outcome certain.

The manager assured us that backups would ensure continuity.

capital / capitol

Capital can mean a city, wealth, or uppercase letter.

Capitol is a legislative building.

The state capital built a new capitol.

personal / personnel

Personal means private or individual.

Personnel means staff or employees.

The manager discussed a personal concern with the personnel director.

precedent / precedence

Precedent is an earlier example.

Precedence means priority in order or importance.

The ruling set a precedent; safety takes precedence over speed.

systemic / systematic

Systemic means affecting an entire system.

Systematic means methodical.

The systematic review revealed a systemic funding problem.

AYLOTI • Sir Hunain Zia • SAT Score 1594/1600 • 154 Total A Grades (Personal), 11 World Records, 7 Distinctions

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Reading & Writing

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Transitions and Logical Flow

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Your Digital SAT Launch Checklist

Central Ideas: Find the Author’s Real Point

Central Ideas: Find the Author’s Real Point

Central Ideas: Find the Author’s Real Point

Central Ideas: Find the Author’s Real Point

Central Ideas: Find the Author’s Real Point

Central Ideas: Find the Author’s Real Point

Central Ideas: Find the Author’s Real Point

Central Ideas: Find the Author’s Real Point

Key Details: Locate, Translate, Verify

Key Details: Locate, Translate, Verify

Key Details: Locate, Translate, Verify

Key Details: Locate, Translate, Verify

Key Details: Locate, Translate, Verify

Key Details: Locate, Translate, Verify

Key Details: Locate, Translate, Verify

Key Details: Locate, Translate, Verify

Key Details: Locate, Translate, Verify

Inferences: The Must-Be-True Standard

Inferences: The Must-Be-True Standard

Inferences: The Must-Be-True Standard

Inferences: The Must-Be-True Standard

Inferences: The Must-Be-True Standard

Inferences: The Must-Be-True Standard

Inferences: The Must-Be-True Standard

Inferences: The Must-Be-True Standard

Inferences: The Must-Be-True Standard

Inferences: The Must-Be-True Standard

Command of Textual Evidence

Command of Textual Evidence

Command of Textual Evidence

Command of Textual Evidence

Command of Textual Evidence

Command of Textual Evidence

Command of Textual Evidence

Command of Textual Evidence

Command of Textual Evidence

Command of Textual Evidence

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Quantitative Evidence: Tables and Graphs

Claims, Findings, and Valid Conclusions

Claims, Findings, and Valid Conclusions

Claims, Findings, and Valid Conclusions

Claims, Findings, and Valid Conclusions

Claims, Findings, and Valid Conclusions

Claims, Findings, and Valid Conclusions

Claims, Findings, and Valid Conclusions

Claims, Findings, and Valid Conclusions

Claims, Findings, and Valid Conclusions