Quick Examples

Here are some sample problems with their solutions in each of the topical areas.

Pre-Algebra

Prime Factorization

 What is the prime factorization of 340?

• 2×5×17²
• 2³×5×17²
• 340
• 2×5²×17
• ✓2²×5×17

 Solution:

The prime factorization of a number is found by determining which prime numbers the number is divisible by. We’ll start by trying to divide it by 2, and if we’re not able to, we try to divide it by increasingly larger prime numbers. We repeat this with the number that is left until we’re left with a prime number.

340 is divisible by 2.

340/2 = 170

2×170 = 340

170 is divisible by 2.

170/2=85

2×2×85 = 340

85 is divisible by 5.

85/5 = 17

2×2×5×17 = 340

17 is a prime number so we can’t factor this any further.

 

Least Common Multiple

What is the least common multiple of 40 and 56?

• 2240
• 112
• ✓280
• 560
• 140

 Solution:

The least common multiple is the smallest number which is a multiple of both 40 and 56. To determine the least common multiple we’ll first get the prime factorization for each number.

40 = 2×2×2×5

56 = 2×2×2×7

We’ll now multiply together these factors, but the factors that are found in both numbers we’ll only multiply once.

2×2×2×5×7 = 280

 

Greatest Common Factor

What is the greatest common factor of 78 and 195?

• 15
• 6
• 26
• 13
• ✓39

 Solution:

The greatest common factor (or greatest common divisor) is the largest number which divides both 78 and 195. To determine the greatest common factor we’ll first get the prime factorization for each number.

78 = 2×3×13

195 = 3×5×13

We’ll now take the factors that these two numbers have in common to get the greatest common factor.

3×13 = 39

 

Number Sequence Pattern

In the sequence of numbers, 2, 8, 32, 128, 512, what is the next number?

• 2560
• ✓2048
• 1638
• 2355
• 1741

 Solution:

Each number in this sequence is the product of the previous number and 4. So the next number will be 512 multiplied by 4, or 2048.

 

Unit Conversion

If something is going at a speed of 89 feet per hour, how many inches per day is it traveling?

• 10,253
• 15,379
• ✓25,632
• 64,080
• 6,408

 Solution:

We’ll convert feet per hour into inches per hour by multiplying the number of feet per hour by 12.

89×12 = 1,068

We’ll convert inches per hour into inches per day by multiplying the number of inches per hour by 24.

1,068×24 = 25,632

 

Mean, Mode, Median

Of the numbers 45, 28, 52, 56, 50, 63, 45, 85, what is the mean?

• 52
• 63
• 51
• 45
• ✓53

 Solution:

The mode is the value that appears the most often. The median is the value which is in the middle – as many numbers in the list are less than the median as are greater. This problem calls for the mean. The mean is the average of the numbers, so we’ll divide the sum of the numbers by the number of numbers.

(45+28+52+56+50+63+45+85)/8 = 424/8 = 53

 

Percent Increase/Decrease

What percent increase is it from 75 to 138?

• 184
• 45.7
• 54.3
• 94
• ✓84

 Solution:

We’ll use the formula for determining the percent increase, which is the difference of the two numbers divided by the lower number.

(138−75)/75 = 63/75 = 0.84

We’ll then multiply it by 100 to convert it into percent form.

0.84×100 = 84

 

Fractions

What number is ¹/₃ of the way from 3 ²/₅ to 7 ²/₃?

• 3 8/15
• 4 44/45
• 3 4/15
• 5  13/15
• ✓4 37/45

Solution:

We’ll first determine the difference between the two numbers.

7 2/3 – 3 2/5 = 23/3 – 17/5 = 115/15 – 51/15 = 64/15

Now, we’ll multiply the difference by ¹/₃.

64/15 × 1/3 = 64/45

Now, we’ll add this to the first number.

17/5 + 64/45 = 153/45 + 64/45 = 217/45 = 4 37/45

 

Scientific Notation

Simplify and give the answer in scientific notation: (8.4×10⁴)(6.3×10⁻⁶) / (9×10³)

• 58.8×10¹
• 5.88×10⁷
•  0.588×10⁻⁵
• ✓5.88×10⁻⁵
• 0.588×10⁷

Solution:

We’ll first multiply the expressions in the numerator.

(8.4×10⁴)(6.3×10⁻⁶) / (9×10³) =

(8.4×6.3) × 10⁴⁻⁶ / (9×10³) =

(52.92 × 10⁻²) / (9×10³)

Now we’ll divide.

(52.92×10⁻²) / (9×10³) = (52.92/9)×10⁻²⁻³ = 5.88×10⁻⁵

Since the coefficient is a single digit number to the left of the decimal, then this number is in proper scientific notation.

 

Arithmetic Word Problems

Use the formula, y = a(1 + r)^x, to solve this problem, where a is the initial number, r is the growth rate, and x is the number of time intervals. A certain kind of bacteria doubles every 40 minutes. If a bacteria culture initially contained 1,200 bacteria, how much bacteria exists after 4 hours?

• 38,400
• 19,200
• 153,600
• 307,200
• ✓76,800

 Solution:

We’ll put in the initial number of bacteria for a, and since the bacteria doubles in the time period, the growth rate is set to 1. To determine the number of intervals, we’ll need to divide the total number of hours by the amount of time it takes for the bacteria to double.

y = 1,200(1+1.00)^4/(2/3)

y = 1,200(2⁶)

y = 1,200(64)

y = 76,800

 

Factorials

Use this formula, y!/(x!(y−x)!), where x is equal to the number of people we’re selecting and y is equal to the number of people in the group, to solve this problem: How many different combinations of 5 people can be taken from a group of 14?

• 4,004
• 1,001
• 3,003
• 70
• ✓2,002

 Solution:

To calculate the number of combinations of x people from a group of y people, we need to use the following formula:

y!/(x!(y−x)!)

So we’ll plug in the number of people we’re selecting for x and the number of people in the group for y.

14!/(5!(14−5)!) =

14!/(5!9!) =

(14×13×12×11×10)/(5×4×3×2×1) =

240,240/120 = /2,002

 

Permutations, Combinations

A pizza can be ordered with 1 type of 4 different kinds of crusts, 1 of 2 different kinds of cheeses, and 1 of 5 toppings. How many different kinds of pizza can be ordered?

• 11
• 121
• 45
• 20
• ✓40

 Solution:

The total number of possible outcomes can be determined by multiplying the number of options in each group together. So we’ll multiply together the number of kinds of crusts, the number of kinds of cheeses, and the number of toppings.

4×2×5 = 40

 

Probability

When a die is rolled, what is the probability that it will not turn up 1, 2, 3, or 5?

• 5/6
• ✓1/3
• 2/3
• 1/6
• 1/2

 Solution:

We can determine the probability of one of 4 numbers not being rolled by subtracting 4 from the possible number of outcomes and then dividing it by the possible number of outcomes.

(6−4)/6 = 2/6 = 1/3

Variable Substitution

What is the value of 

 (5x)^1/2 + (8y)^1/2 

 if x = 20 and y = 32?

• 33
• 20
• 16
• 13
• ✓26

 Solution:

We’ll replace x and y with the appropriate values and calculate the expression.

(5x)^1/2 + (8y)^1/2 =

(5×20)^1/2 + (8×32)^1/2 =

(100)^1/2 + (256)^1/2 =

√100 + √256 =

10 + 16 = 26

Radicals – Adding

Add: 10√2 + 16√32

•  26√32
• ✓74√2
• 26√34
• 30√32
• 58√2

 Solution:

When adding radicals, the numbers underneath the radical sign are not added, and they must be equal in order for the numbers outside the radical to be added. Since these radicals are different, we can’t add them so we’ll need to try to simplify them. The first radical can’t be simplified so we’ll try to simplify the second one by finding the largest perfect square that divides evenly into 32.

16√32 = 16×√16×√2 = 16×4×√2 = 64√2

Now we can add the radicals together.

10√2 + 64√2 = (10 + 64)√2 = 74√2

Radicals – Multiplying

Multiply: 6√42 × 12√(5/7)

• 18√7
• 72√42
• ✓72√30
• 102
• 102√30

 Solution:

We’ll multiply the numbers outside the radicals and then the numbers inside the radicals and then simplify.

6√42 × 12√(5/7) =

6×12×√42 × √(5/7) = 72√30

 

Expressions – Exponents

Calculate: 

(18⁴×9⁻⁴) + 256^3/2 + 1,000^4/3 − 4,096^3/4

• 4,080
• 23,800
• ✓13,600
• 46,240
• 17,000

Solution:

We’ll first convert each number and its exponent.

(18⁴×9⁻⁴) + 256^3/2 + 1,000^4/3 − 4,096^3/4 =

(104,976×1/9⁴) + (√256)³ + (√³1000)⁴ − (√⁴4096)³ =

104,976/6561 + 16³ + 10⁴ – 8³ =

16 + 4096 + 10,000 – 512 = 13,600

 

Elementary Algebra

Least Common Multiple

What is the least common multiple of 

14x²y²z and 20x²y²z³?

• 280x²y²z
• 14x²y²z³
• 280x²y²z³
• 20x²y²z
• ✓140x²y²z³

Solution:

The least common multiple is the smallest number which is a multiple of both 14x²y²z and 20x²y²z³. To determine the least common multiple we’ll first get the prime factorization of each expression.

14x²y²z = 2 × 7 × x² × y² × z

20x²y²z³ = 2² × 5 × x² × y² × z³

We’ll now take out the factors that have the highest exponents from each expression and then multiply them together to get the answer.

2² × 5 × 7 × x² × y² × z³ = 140x²y²z³

Greatest Common Factor

What is the greatest common factor of 

54x⁴y²z³ and 84x⁴y³z⁴?

• 3x⁴y³z³
• ✓6x⁴y²z³
• 2x⁴y²z³
• 6x⁴y³z⁴
• 3x⁴y³z⁴

 Solution:

The greatest common factor (or greatest common divisor) is the largest number which is a factor of both 54x⁴y²z³ and 84x⁴y³z⁴. To determine the greatest common factor we’ll first get the prime factorization of each expression.

54x⁴y²z³ = 2×3×3×3 × (x×x×x×x) × y×y × z×z×z

84x⁴y³z⁴ = 2×2×3×7 × (x×x×x×x) × (y×y×y) × (z×z×z×z)

We’ll now take out the factors these two expressions have in common and multiply them together to get the answer.

2×3 × (x×x×x×x) × y×y × z×z×z = 6x⁴y²z³

Expressions – Simplify

Simplify this expression: 

(6x⁵+18x⁴−108x³)/(7x³+28x²−147x)

• ✓6x²(x+6)/7(x+7)
• 6x²(x−6)/7(x−7)
• 6x²(x+7)/7(x−9)
• 6x²(x−2)/7(x+3)
• 6x²(x−9)/7(x+21)

Solution:

We simplify by breaking the expression down into factors. We’ll first take a factor out of the expressions in the numerator and the denominator.

(6x⁵ + 18x⁴ − 108x³)/(7x³ + 28x² − 147x) = 6x³(x² + 3x − 18)/7x(x² + 4x − 21)

We can now factor what’s inside the parentheses. We’ll factor the numerator by determining which product and sum of two numbers equal -18 and 3, respectively, and we’ll factor the denominator by determining which product and sum of two numbers equal -21 and 4, respectively. Those two numbers are -3 and 6 for the numerator and -3 and 7 for the denominator.

6x³(x + 6)(x − 3)/7x(x − 3)(x + 7) = 6x²(x + 6)/7(x + 7)

Expressions – Factoring

Which of the following is a correct factoring of this expression:

10x⁴ + 125x³ + 360x²?

• ✓5x²(2x + 9)(x + 8)
• (5x² + 18x)(2x² + 20x)
• 5x⁴(2x + 18)(x + 4)
• (5x² + 12x)(2x² + 30x)
•  5x³(2x + 36)(x + 2)

 Solution:

To factor this expression, we’ll first take out the greatest common factor to make it easier to work with.

10x⁴ + 125x³ + 360x² = 5x²(2x² + 25x + 72)

We’ll now factor what’s in the parentheses. We can factor it by determining which two numbers have a product of 72 and when multiplied by the factors of 2 have a sum of 25. Those two numbers are 8 multiplied by 2 and 9 multiplied by 1.

5x²(2x + 9)(x + 8)

By using the FOIL method, we can check to see if we factored this correctly. We’ll multiply the First parts of the two polynomials, then the Outside parts, then the Inside parts, and then the Last parts.

5x²((2x × x) + (2x × 8) + (9 × x) + (9 × 8)) =

5x²(2×2 + 16x + 9x + 72) =

5x²(2x² + 25x + 72) =

10x⁴ + 125x³ + 360x²

Expressions – Exponents

Simplify: (x⁻²y⁵z⁻⁵)⁵

• ✓y²⁵/x¹⁰z²⁵
• y²⁵x²z⁵
• x³y²⁵z⁵
• y¹⁰x²
• y¹⁰x²z²⁵

 Solution:

To simplify this expression, we’ll multiply the exponent of each variable by the exponent of the entire expression.

(x⁻²y⁵z⁻⁵)⁵ =

x^{−2 × 5} y^{5 × 5} z^{−5 × 5} =

x⁻¹⁰y²⁵z⁻²⁵

To further simplify this, we need to get rid of the negative exponents. A negative exponent is equal to the reciprocal of the variable raised to the positive form of the exponent.

(1/x¹⁰) y²⁵ (1/z²⁵) = y²⁵/x¹⁰ z²⁵

Radicals – Simplifying

Simplify: 16√1575

• 32√2
• 128√5
• 80√7
• ✓240√7
• 208√2

 Solution:

We’ll solve this by finding the largest perfect square that divides evenly into 1575 and then multiply it by 16.

16√1575 =

16 × √1575 =

16 × √225 × √7 =

16×15×√7 = 240√7

Equations – Algebra

Find the value of x in this equation: 

(8x + 16)/12 = (4x – 40)/2

• ✓16
• −2
• 9
• 11
• 8

 Solution:

(8x + 16)/12 = (4x − 40)/2

12((8x + 16)/12) = 12((4x − 40)/2)

8x + 16 = 6(4x − 40)

8x + 16 = 24x − 240

16 + 240 = 24x − 8x

256 = 16x

256/16 = x

16 = x

Equations – Word Problems

A baseball player’s slugging percentage is calculated by adding up the number of bases he gets for each hit (1 for a single, 2 for a double, 3 for a triple, and 4 for a home run) and dividing that by the number of at-bats he has. If a player has a slugging percentage of .375 and 120 total bases, how many at-bats has that player had?

• 300
• 345
• 360
• ✓320
• 275

 Solution:

To figure out the number of at-bats this player has had, set up an equation with x equaling the number of at-bats.

.375 = 120/x

Solve for x.

Multiply both sides by x.

.375x = 120

Divide both sides by .375.

x = 120/.375 = 320

Variation

If y varies directly as x and inversely as z, what is the value of z when x = 28 and y = 4 if x = 2 when y = 32 and z = 12?

• 336
• 3226
• ✓1344
• 4570
• 2016

Solution:

Since we’re dealing with a combination variation problem, we’ll combine the formula for direct variation, y = kx, with the formula for inverse variation, y = ^k/_x. So the formula becomes y = ^kx/_z. We’ll use this first to determine the constant of variation.

32 = 2k/12 = k/6

Solve for k.

32×6 = k

192 = k

192 is the constant of variation, so we’ll plug this value for k into the formula along with the values of x and y given in the question to determine the value of z.

4 = (192)(28)/z

Solve for z.

4z = 5376

z = 5376/4 = 1344

Intermediate Algebra

Equations – Algebra

Find the value of x in this equation: 

x² − 26x – 31 = 0

• 3±6√19
• 5±4√11
• 13±2√3
• ✓13±10√2
• 14±6√15

 Solution:

We’re not able to factor this equation so we’ll solve it by completing the square. We’ll take the constant value and put it on the other side of the equation. We’ll then take half of the coefficient of x and square it, and then add it to both sides of the equation. We’ll then be able to factor it and solve for x.

x² − 26x – 31 = 0

x² − 26x = 0+31 = 31

x²−26x+(26/2)²=31+(26/2)²

x² − 26x + 169 = 200

(x − 13)² = 200

We’ll now take the square root of each side. We’ll need to use the plus-minus sign when we take the square root of a number since both the positive and negative values, when squared, will give us the original number.

√(x – 13)² = ±√200

x − 13 = ±10√2

x = 13±10√2

Equations – Functions

If f(x) = −3x − 21 and g(x) = −3x − 10, then what is 

 f(g(−3))?

• ✓−18
• 24
• 13
• −1
• 31

 Solution:

We’ll start with the inside part of the question and work our way out. We need to first determine what g(-3) is, so we’ll substitute the -3 for x in the g(x) function.

−3(−3) – 10 = 9 – 10 = −1

We’ll now substitute the -1 for x in the f(x) function.

−3(−1) – 21 = 3 – 21 = −18

Two Variable Word Problems

Larry is planning two road trips. The first is 137 miles longer than the second, and the two put together are 1349 miles. How long is the second trip?

• ✓606
• 594
• 568
• 618
• 582

 Solution:

To figure out the length of the second road trip, we’ll form two equations with what we know about the trips, with x equaling the length of the first road trip and y equaling the length of the second road trip.

x + y = 1349

x – y = 137

We’ll add these two equations together and solve for x.

x + y + x – y = 1349 + 137

2x = 1486

x = 1486/2 = 743

Since x is equal to the length of the first road trip, we’ll put this value in for x in the equations that we formed and solve for y.

743 + y = 1349

y = 1349 – 743 = 606

Logarithms

What is the value of x in this equation: 

 log₃x = −3?

• 1/81
• 1/9
• ✓1/27
• −9
• −1

 Solution:

A logarithm (-3 in this problem) is a number that represents the power to which a number (called the base, 3 in this problem) must be raised to get another number (x in this problem). The value of x in this equation is equal to 3^-3, so we can calculate this to determine what x is.

3⁻³ = 1/3³ = 1/27

Complex Numbers

What is the value of x in this equation: 

x² − 4x + 16 = 0?

• 2±2√3
• 2i√3
• ✓2±2i√3
• {2,8}
• 4±2i√3

 Solution:

We’re not able to factor this equation so we’ll solve it by completing the square. To complete the square, we’ll take the constant value and put it on the other side of the equation. We’ll then take half of the coefficient of x and square it, and then add it to both sides of the equation. We’ll then be able to factor it and solve for x. When the square root of a value is found, we need to put the ± symbol in front of it since both the positive and negative squared value of the square root will result in the same value. Also, in this problem we’ll be working with the square root of a negative number which is expressed as an imaginary number, or i.

x² − 4x + 16 = 0

x² − 4x = 0 − 16

x² − 4x + (4/2)² = −16 + (4/2)²

x² − 4x + 4 = −16 + 4 = −12

(x−2)² = −12

√(x−2)² = √(−12)

x−2 =  ±√(−12) = ±√−1 × √12 = ±i√12

x = 2±2i√3

Inequalities

Solve this inequality for x: 

13 − 8x < −29 − x

• x < 6
• x > 7
• ✓x > 6
• x > −6
• x < 7

 Solution:

Solve for x.

Inequalities are solved just like equations, except that we change the direction of the inequality if we multiply or divide by a negative number.

13 − 8x < −29 − x

−8x + x < −29 − 13

−7x < −42

x > −42/−7

We switched the sign since we divided by a negative number.

x > 6

Test the answer by picking a number greater than 6 and insert it in for x in the inequality and calculate it to determine if the inequality is true.

Coordinate Geometry

Lines

What is the equation of a line that passes through the point (-2,3) and is parallel to the line given by the equation 9x + 3y = 6?

• −1/3x + 2 = y
• ✓−3x – 3 = y
• 1/3x + 3 = y
• 1/3x – 3 = y
• 3x + 2 = y

 Solution:

We’ll need to convert the given equation into slope-intercept form, y = mx + b, with m being the slope and b being the y-intercept.

9x + 3y = 6

9x – 6 = −3y

(9x – 6)/−3 = y

−3x + 2 = y

We can use this slope for the equation of the line that we’re trying to find since parallel lines have equal slopes. To find the y-intercept, we’ll use the form, y – y₁ = m(x – x₁), where we use the coordinates of the point that is given for x₁ and y₁.

y – 3 = −3(x + 2)

y – 3 = −3x − 6

y = −3x − 6 + 3

y = −3x − 3

Shape Coordinates

Point A (-7,-2), point B (7,-6), point C (-5,5), and point D form a rectangle in the standard coordinate plane. What are the coordinates of point D?

• (10, 1)
• ✓(9, 1)
• (9, 2)
• (8, 1)
• (8, 2)

 Solution:

First, plot the coordinates to get an idea of what the rectangle looks like. Then, you can observe that the x-coordinate of point A is 14 less than the x-coordinate of point B, and the y-coordinate of point A is 4 greater than the y-coordinate of point B. If you add 14 to the x-coordinate of point C and subtract 4 from the y-coordinate of point C, then you will get the coordinates of point D, (9,1).

Circles

What is the radius of a circle represented by this equation: 

x² + y² + 4x − 18y – 59 = 0

• 2
• 4
• ✓12
• 9
• 18

 Solution:

We’ll need to convert this equation into the standard equation for a circle which is

 (x − h)² + (y − k)² = r²

where (h,k) is the midpoint of the circle and r is the radius. We can figure this out by completing the square for both the x variable and the y variable. We’ll take the coefficients of x and y, divide them in half, and then square them. We’ll also move the integer over to the other side of the equation and add to it the square of the numbers we get when we complete the square.

x² + y² + 4x − 18y – 59 = 0

(x + 4/2)²+(y – 18/2)²= 59 + 4 + 81

(x + 2)² + (y − 9)² = 144

The radius is the square root of 144, or 12.

Plane Geometry

Circles

A rectangle with a width of 14 and a length of 15 has a circle inside of it that is tangent to two opposite sides of the rectangle. If a point is chosen randomly from inside the rectangle, what is the probability that the point will be inside the circle?

• π/15
• 7π/29
• ✓7π/30
• 29π/98
• 14/15π

 Solution:

To solve this problem, we’ll take the ratio of the area of the circle to the area of the rectangle.

πr²/(14×15) = πr²/210

Because the circle is tangent with two opposite sides of the rectangle, then we can determine that the radius is equal to half of the width of the rectangle, or 7. We’ll put 7 in for the radius.

π7²/210 = 49π/210 =7π/30

Area

A rectangle has an area of 408 square inches. If you multiply each side of the rectangle by 2, what will the area of the rectangle be?

• 2,285
• 1,306
• 816
• 979
• ✓1,632

Solution:

We can determine the new area of the rectangle by multiplying the area by (2)².

408 × (2)² =

408 × (2 × 2) =

408 × 4 = 1,632

Angles

Line AB is parallel to Line CD. EF intersects AB at point G and CD at point H. The value of ∠EHD is 7x + 1, and the value of ∠EGA is 3x – 11. What is the size of ∠EGA?

• 224°
• ✓46°
• 19°
• 134°
• 314°

 Solution:

When we draw the lines out and label the points and the angles, we’ll see that because lines AB and CD are parallel, the sum of the angles in the problem will be 180°. So, to determine the value of x, we can add the two angles together and set them equal 180°.

7x + 1 + 3x – 11 = 180

Solve for x.

10x − 10 = 180

10x = 180 + 10 = 190

x = 190/10 = 19

We’ll now plug this value of x in to get the value of angle EGA.

3x − 11 = (3×19) − 11 = 46

Pythagorean Theorem

The perimeter of a right triangle is 96 inches. If the hypotenuse is 40 inches, how many square inches is the area of the triangle?

• 374
• 392
• 390
• ✓384
• 360

 Solution:

To solve this problem, we’ll set up an equation with x equaling the length of one of the other two sides and y equaling the other side.

x + y = 96 – 40 = 56

Solve for y.

y = 56 − x

We’ll now plug this value into the Pythagorean Theorem, a² + b²= c², with a and b being the two sides of the triangle we don’t know and c being the hypotenuse. We’ll set a equal to x and b equal to y.

x² + (56 − x)² = 40²

x² + (56 − x)(56 − x) = 1600

x² + 3136 − 56x − 56x + x² = 1600

2x² − 112x + 3136 – 1600 = 0

2x² − 112x + 1536 = 0

2(x² − 56x + 768) = 0

2(x − 24)(x − 32) = 0

The two possible values for x are 24 and 32. So, these are the lengths of the other two sides. Since these two sides make up the base and height of the right triangle, we can figure out the area of the triangle by plugging these values into the formula for determining the area of a triangle, which is base times height divided by 2. This equals 384.

Triangles – 30-60-90

In △ABC, ∠ABC is a right angle, ∠BCA is 30°, and BC is 29√3 inches. How many inches is the length of AB?

• 29√3
• 70
• ✓29
• 58
• 70√3

 Solution:

Draw out the triangle with all of the sides and angles labeled to get a better idea of what is being asked. This is a 30° – 60° – 90° triangle, so we can determine the length of the other sides if we know the length of one side. If the short side is x, then the longer side will be x√3, and the hypotenuse is 2x. We need to determine what side the question is giving and which side it’s asking for. The given side is opposite the 60° angle, so it is the longer side. The side in question is opposite the 30° angle, so it is the shorter side. We can calculate the length of the shorter leg by dividing the longer leg by √3.

(29√3) / √3 = 29

Triangles – 45-45-90

In an isosceles right triangle, the length of one of the legs is 19√2 inches. How many inches long is the hypotenuse?

• 19√6
• 38√2
• 19√6/3
• ✓38
• 19

 Solution:

To determine the length of the hypotenuse of a 45° – 45° – 90° triangle, we multiply the length of one of the legs by √2.

19√2 × √2 = 38

Trigonometry

In △ABC, ∠ABC is a right angle. ∠BAC is 28.1°, and AC is 34 inches. How many inches is the length of BC, rounded to the nearest inch?

• 19
• ✓16
• 37
• 12
• 30

 Solution:

First, draw the triangle out with the sides and angles labeled so we can get a better idea of what is being asked. Since the side that is being asked for is opposite the angle that is given and the given length of the side of the triangle is the hypotenuse, we know that we are working with the sine of the given angle. So, we’ll set up an equation with the sine of the angle set equal to the opposite divided by the hypotenuse, with the opposite set to x.

 

sin 28.1 = x/34

0.471 = x/34

0.471×34 = x

16 ≈ x