Precalculus Quiz: Solving Right Triangles Pythagorean Theorem Trigonometry

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Solving Right Triangles Pythagorean Theorem TrigonometryQuestion 1 of 20

A right triangle has legs of length $a$ and $b$ , with hypotenuse $c$ . If one acute angle measures $\theta$ , and $\sin(\theta) = \frac{3}{5}$ , which expression correctly represents $\cos(90° - \theta)$ in terms of the triangle's sides?

\cos(90° - \theta) = \frac{\text{adjacent to } \theta}{\text{hypotenuse}} = \frac{4}{5}

\cos(90° - \theta) = \sin(\theta) = \frac{\text{opposite to } \theta}{\text{hypotenuse}} = \frac{3}{5}

\cos(90° - \theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite to } \theta} = \frac{5}{3}

\cos(90° - \theta) = \tan(\theta) = \frac{\text{opposite to } \theta}{\text{adjacent to } \theta} = \frac{3}{4}

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Precalculus Quiz

Precalculus Quiz: Solving Right Triangles Pythagorean Theorem Trigonometry

Practice Solving Right Triangles Pythagorean Theorem Trigonometry in Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Solving Right Triangles Pythagorean Theorem Trigonometry, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$\cos(90° - \theta) = \frac{\text{adjacent to } \theta}{\text{hypotenuse}} = \frac{4}{5}$
$\cos(90° - \theta) = \sin(\theta) = \frac{\text{opposite to } \theta}{\text{hypotenuse}} = \frac{3}{5}$ (correct answer)
$\cos(90° - \theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite to } \theta} = \frac{5}{3}$
$\cos(90° - \theta) = \tan(\theta) = \frac{\text{opposite to } \theta}{\text{adjacent to } \theta} = \frac{3}{4}$

Explanation: For complementary angles, cos(90° - θ) = sin(θ). Since sin(θ) = 3/5, we have cos(90° - θ) = 3/5. This represents the ratio of the side opposite to θ over the hypotenuse. Choice A incorrectly applies cos(θ) = 4/5 instead of the complementary relationship. Choice C confuses the relationship with the reciprocal (cosecant). Choice D incorrectly uses the tangent ratio instead of recognizing the complementary angle relationship.

Question 2

A ladder is $10$ ft long and leans against a vertical wall. The bottom of the ladder is $6$ ft from the wall, forming a right triangle with the ground and the wall. What is the height (in feet) the ladder reaches up the wall?

$4$ ft
$16$ ft
$8$ ft (correct answer)
$\sqrt{136}$ ft

Explanation: This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². The ladder forms the hypotenuse (10 ft), the distance from the wall is one leg (6 ft), and we need to find the height up the wall, which is the other leg, so we use: 6² + height² = 10², which gives 36 + height² = 100, so height² = 64, and taking the square root yields height = 8 ft. Choice C is correct because when we substitute the given values into the rearranged Pythagorean theorem, we get height = √(10² - 6²) = √(100 - 36) = √64 = 8 ft. Choice A incorrectly subtracts the sides instead of using the Pythagorean theorem: 10 - 6 = 4 instead of √(10² - 6²) = 8. This is a 6-8-10 triangle (a scaled 3-4-5 triple), so we can recognize immediately that the missing side is 8 without calculation. Recognize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples to save time on calculations—if you see two sides of a triple, the third can be determined without calculation.

Question 3

A right triangle has one leg of length 7 and hypotenuse of length 25. An angle $\theta$ in this triangle satisfies $\cos(\theta) = \frac{7}{25}$ . What is the value of $\sin(90° - \theta) + \cos(90° - \theta)$ ?

$\sin(90° - \theta) + \cos(90° - \theta) = \frac{7}{25} + \frac{24}{25} = \frac{31}{25}$ (correct answer)
$\sin(90° - \theta) + \cos(90° - \theta) = \frac{24}{25} + \frac{7}{25} = \frac{31}{25}$
$\sin(90° - \theta) + \cos(90° - \theta) = \frac{24}{25} + \frac{24}{25} = \frac{48}{25}$
$\sin(90° - \theta) + \cos(90° - \theta) = \frac{7}{25} + \frac{7}{25} = \frac{14}{25}$

Explanation: First, find the other leg using the Pythagorean theorem: other leg = √(25² - 7²) = √(625 - 49) = √576 = 24. Given cos(θ) = 7/25, we can find sin(θ) = 24/25. Using complementary angle relationships: sin(90° - θ) = cos(θ) = 7/25 and cos(90° - θ) = sin(θ) = 24/25. Therefore, sin(90° - θ) + cos(90° - θ) = 7/25 + 24/25 = 31/25. Choice B reverses the order but gets the same sum. Choice C incorrectly uses sin(θ) for both terms. Choice D incorrectly uses cos(θ) for both terms.

Question 4

In right triangle $\triangle MNO$ , $\angle O = 90^\circ$ , the hypotenuse is $MN = 12$ cm, and $\angle M = 30^\circ$ . Using the given information, what is the exact length of side $NO$ (the side opposite $\angle M$ )?

$6$ cm (correct answer)
$6\sqrt{3}$ cm
$12\sqrt{3}$ cm
$4\sqrt{3}$ cm

Explanation: This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. From the perspective of angle M (30°), the side NO is opposite, and MN=12 is the hypotenuse, so we use sin(M) = NO/12; since sin(30°)=1/2, NO=12 × (1/2)=6. Choice A is correct because it uses the sine ratio for the 30° angle: 12 sin(30°) = 12 × (1/2) = 6 cm. Choice B confuses the 30-60-90 triangle ratios with 45-45-90 triangle ratios, using √3 where it's not needed for the side opposite 30°. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. For complementary angles in a right triangle, sin(θ) = cos(90° - θ), which explains why the sine of one acute angle equals the cosine of the other.

Question 5

In right triangle $\triangle STU$ , $\angle U = 90^\circ$ , $SU = 12$ (leg adjacent to $\angle S$ ), and $TU = 5$ (leg opposite $\angle S$ ). Using the given information, what is the value of $\tan(\angle S)$ ?

$\frac{12}{5}$
$\frac{5}{13}$
$\frac{5}{12}$ (correct answer)
$\frac{13}{5}$

Explanation: This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. From the perspective of angle S, the side TU=5 is opposite, and SU=12 is adjacent, so we use tan(S) = opposite/adjacent = 5/12. Choice C is correct because it uses the tangent ratio with the opposite and adjacent sides: tan(S) = 5/12. Choice A inverts the ratio, calculating adjacent/opposite = 12/5 instead of opposite/adjacent. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known).

Question 6

A ladder leans against a wall, making an angle $\alpha$ with the ground. The ladder's length is 20 feet, and it reaches 16 feet up the wall. Using the relationship between sine and cosine of complementary angles, what is $\cos(\alpha) + \sin(90° - \alpha)$ ?

$\cos(\alpha) + \sin(90° - \alpha) = \frac{16}{20} + \frac{12}{20} = \frac{28}{20} = 1.4$
$\cos(\alpha) + \sin(90° - \alpha) = \frac{16}{20} + \frac{16}{20} = \frac{32}{20} = 1.6$
$\cos(\alpha) + \sin(90° - \alpha) = \frac{12}{20} + \frac{16}{20} = \frac{28}{20} = 1.4$
$\cos(\alpha) + \sin(90° - \alpha) = \frac{12}{20} + \frac{12}{20} = \frac{24}{20} = 1.2$ (correct answer)

Explanation: When you encounter a ladder problem with trigonometry, you're working with a right triangle where the ladder is the hypotenuse, the wall height is the opposite side to angle α, and the ground distance is the adjacent side to angle α. First, let's find the missing side using the Pythagorean theorem. With a 20-foot ladder reaching 16 feet up the wall, the ground distance is

\sqrt{20^2 - 16^2} = \sqrt{400 - 256} = \sqrt{144} = 12

feet. Now we can find the trigonometric ratios:

\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{16}{20}

and

\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{20}

. The key insight is understanding complementary angle relationships. When two angles are complementary (sum to 90°), the sine of one equals the cosine of the other. Therefore,

\sin(90° - \alpha) = \cos(\alpha) = \frac{12}{20}

. So

\cos(\alpha) + \sin(90° - \alpha) = \frac{12}{20} + \frac{12}{20} = \frac{24}{20} = 1.2

, which is answer D. Answer A incorrectly uses

\sin(\alpha)

for

\cos(\alpha)

, confusing opposite and adjacent sides. Answer B makes both errors—wrong values for both terms. Answer C correctly identifies

\cos(\alpha) = \frac{12}{20}

but incorrectly uses

\sin(\alpha) = \frac{16}{20}

instead of applying the complementary angle relationship. Remember: for complementary angles,

\sin(90° - \theta) = \cos(\theta)

and

\cos(90° - \theta) = \sin(\theta)

. This relationship is fundamental in trigonometry and appears frequently on exams.

Question 7

In right triangle $JKL$ , the right angle is at $K$ . If $JK = 9$ and $KL = 12$ , what is the exact length of the hypotenuse $JL$ ?

$15$ (correct answer)
$21$
$\sqrt{369}$
$\sqrt{81} + \sqrt{144}$

Explanation: This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Given sides of length 9 and 12, we substitute into the Pythagorean theorem: 9² + 12² = c², which gives 81 + 144 = c², so c² = 225, and taking the square root yields c = 15. Choice A is correct because it shows correct substitution into the Pythagorean theorem with specific numbers 9 and 12, yielding √(81 + 144) = √225 = 15. Choice D makes the error of adding the square roots instead of taking the square root after adding the squares: √81 + √144 = 9 + 12 instead of √(81 + 144) = 15. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). Recognize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples to save time on calculations—if you see two sides of a triple, the third can be determined without calculation.

Question 8

In right triangle $ABC$ , the right angle is at $C$ . The hypotenuse is $AB=13$ m and one leg is $BC=5$ m. What is the exact length of the other leg $AC$ ?

$\sqrt{194}$ m
$12$ m (correct answer)
$8$ m
$18$ m

Explanation: This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Since we know the hypotenuse (13 m) and one leg (BC = 5 m), we need to find the other leg AC, so we rearrange: AC² + 5² = 13², which gives AC² + 25 = 169, so AC² = 144, and taking the square root yields AC = 12. Choice B is correct because this is a 5-12-13 Pythagorean triple, so we can recognize immediately that the missing side is 12 without calculation. Choice A incorrectly adds the squares: √(13² + 5²) = √(169 + 25) = √194, treating both given sides as legs instead of recognizing that 13 is the hypotenuse. When using the Pythagorean theorem, always check that you're putting the hypotenuse (the longest side) as c in the equation a² + b² = c², not one of the legs. Remember that in any right triangle, the hypotenuse is always the longest side, so if you're given two sides, the larger one is the hypotenuse if it's opposite the right angle.

Question 9

In right triangle $ABC$ , the right angle is at $C$ . If $\angle A=60^\circ$ and $AC=8$ (adjacent to $\angle A$ ), which equation can be used to find the hypotenuse $AB$ ?

$\sin(60^\circ)=\dfrac{8}{AB}$
$\cos(60^\circ)=\dfrac{8}{AB}$ (correct answer)
$\tan(60^\circ)=\dfrac{8}{AB}$
$\cos(60^\circ)=\dfrac{AB}{8}$

Explanation: This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. From the perspective of angle A = 60°, the side AC = 8 is adjacent to angle A, and AB is the hypotenuse we need to find, so we use cos(60°) = adjacent/hypotenuse = 8/AB. Choice B is correct because cos(60°) = AC/AB = 8/AB, correctly identifying AC as the adjacent side to angle A and AB as the hypotenuse. Choice A incorrectly uses sine instead of cosine, which would require the opposite side BC rather than the adjacent side AC that we're given. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. When setting up equations to solve for an unknown side, place the unknown in the denominator if it's the hypotenuse or larger side, then cross-multiply to solve.

Question 10

From a point on level ground, a surveyor measures the angle of elevation to the top of a tower as $45^\circ$ . The surveyor is 30 meters from the base of the tower. Using the given information, what is the height of the tower (in meters)?

15 m
30 m (correct answer)
$30\sqrt{2}$ m
$\frac{30}{\sqrt{2}}$ m

Explanation: This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. Since we know the adjacent side is 30 m and the angle is 45°, we use the tangent ratio: tan(45°) = height/30, and since tan(45°) = 1, solving gives height = 30 m. Choice B is correct because it uses the tangent ratio with tan(45°) = 1 and adjacent = 30, yielding height = 30. Choice C confuses the 45-45-90 triangle ratios, incorrectly multiplying by √2 instead of recognizing the equal opposite and adjacent sides at 45°. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known).

Question 11

A surveyor measures the angle of elevation to the top of a building as $32°$ from a point 150 feet away from the base. Using the relationship $\sin(32°) = \cos(58°) = 0.530$ and $\cos(32°) = \sin(58°) = 0.848$ , what is the height of the building?

The height is $150 \cdot \tan(32°) = 150 \cdot \frac{\sin(32°)}{\cos(32°)} = 150 \cdot \frac{0.530}{0.848} \approx 93.8$ feet (correct answer)
The height is $150 \cdot \tan(58°) = 150 \cdot \frac{\sin(58°)}{\cos(58°)} = 150 \cdot \frac{0.848}{0.530} \approx 240.0$ feet
The height is $150 \cdot \cot(32°) = 150 \cdot \frac{\cos(32°)}{\sin(32°)} = 150 \cdot \frac{0.848}{0.530} \approx 240.0$ feet
The height is $150 \div \tan(32°) = 150 \div \frac{\sin(32°)}{\cos(32°)} = 150 \cdot \frac{0.848}{0.530} \approx 240.0$ feet

Explanation: In this right triangle, the horizontal distance is 150 feet (adjacent to the 32° angle) and the height is the opposite side. Using tan(32°) = opposite/adjacent = height/150, we get height = 150 × tan(32°) = 150 × (sin(32°)/cos(32°)) = 150 × (0.530/0.848) ≈ 93.8 feet. Choice B incorrectly uses the complementary angle 58° instead of 32°. Choice C uses cotangent instead of tangent, which would give the horizontal distance if the height were known. Choice D uses the reciprocal of tangent, making the same error as Choice C.

Question 12

A ladder is 13 ft long and leans against a vertical wall. The bottom of the ladder is 5 ft from the wall, forming a right triangle with the ground. What is the height on the wall that the ladder reaches?

$8$ ft
$12$ ft (correct answer)
$18$ ft
$\sqrt{194}$ ft

Explanation: This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². The ladder forms a right triangle where the ladder itself is the hypotenuse (13 ft), the distance from the wall is one leg (5 ft), and the height on the wall is the other leg, so we use: 5² + h² = 13², which gives 25 + h² = 169, so h² = 144, and taking the square root yields h = 12. Choice B is correct because this is a 5-12-13 Pythagorean triple, so we can recognize immediately that the missing side is 12 without calculation. Choice D incorrectly adds the squares: √(13² + 5²) = √(169 + 25) = √194, treating both given measurements as legs instead of recognizing that 13 is the hypotenuse. Recognize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples to save time on calculations—if you see two sides of a triple, the third can be determined without calculation. In real-world problems, identify which measurement represents the hypotenuse (often a ladder, ramp, or diagonal distance) versus the legs (usually horizontal and vertical distances).

Question 13

A ladder is 10 feet long and leans against a vertical wall. The bottom of the ladder is 6 feet from the wall, forming a right triangle with the ground and the wall. What is the height (in feet) the ladder reaches up the wall?

4 ft
8 ft (correct answer)
12 ft
$\sqrt{136}$ ft

Explanation: This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Given hypotenuse of length 10 and leg of length 6, we rearrange to find the other leg: b² = 10² - 6², which gives 100 - 36 = b², so b² = 64, and taking the square root yields b = 8. Choice B is correct because it shows correct substitution into the Pythagorean theorem with 10 as hypotenuse and 6 as one leg, yielding 8. Choice D incorrectly treats 10 and 6 as legs instead of hypotenuse and leg, using √(10² + 6²) = √136 instead of correctly identifying that 10 is the hypotenuse. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). When using the Pythagorean theorem, always check that you're putting the hypotenuse (the longest side) as c in the equation a² + b² = c², not one of the legs.

Question 14

A surveyor stands $24$ m from the base of a vertical tower. The angle of elevation from the surveyor to the top of the tower is $45^\circ$ . Using the given information, what is the height of the tower?

$12$ m
$24$ m (correct answer)
$24\sqrt{2}$ m
$48$ m

Explanation: This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. Since we know the adjacent side (24 m distance to base) and the angle of elevation (45°), and need the opposite side (height of tower), we use tan(45°) = height/24; since tan(45°)=1, height=24 × 1=24. Choice B is correct because it uses the tangent ratio with the given angle and adjacent side: 24 tan(45°) = 24 × 1 = 24 m. Choice C confuses the 45-45-90 triangle ratios, multiplying by √2 as if finding the hypotenuse instead of the opposite side. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known).

Question 15

In right triangle $\triangle PQR$ , $\angle R = 90^\circ$ . The hypotenuse is $PQ = 10$ and the leg adjacent to $\angle P$ is $PR = 8$ . Using the given information, what is the value of $\cos(\angle P)$ ?

$\frac{5}{4}$
$\frac{4}{5}$ (correct answer)
$\frac{3}{5}$
$\frac{5}{3}$

Explanation: This question tests the ability to solve right triangles using trigonometric ratios. Trigonometric ratios relate the angles of a right triangle to the ratios of its sides: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. From the perspective of angle P, the side PR=8 is adjacent, and PQ=10 is the hypotenuse, so we use cos(P) = adjacent/hypotenuse = 8/10 = 4/5. Choice B is correct because it uses the cosine ratio with the adjacent side and hypotenuse: cos(P) = 8/10 = 4/5. Choice C reverses the sine and cosine ratios, using the opposite side instead of adjacent, which swaps the ratios. Remember the SOH-CAH-TOA mnemonic for choosing the correct trig ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent, where opposite and adjacent are always relative to the angle in question. For complementary angles in a right triangle, sin(θ) = cos(90° - θ), which explains why the sine of one acute angle equals the cosine of the other.

Question 16

In right triangle $\triangle GHI$ , $\angle I = 90^\circ$ , $GH = 10$ m (hypotenuse), and $GI = 6$ m. Using the given information, what is the length of side $HI$ ?

$4$ m
$8$ m (correct answer)
$\sqrt{136}$ m
$\sqrt{64}$ m

Explanation: This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Given the hypotenuse of length 10 and one leg of length 6, we rearrange the Pythagorean theorem to find the other leg: HI = √(10² - 6²) = √(100 - 36) = √64 = 8. Choice B is correct because it shows correct substitution into the Pythagorean theorem with the hypotenuse as c and solving for the missing leg: √(100 - 36) = 8 m. Choice C calculates √(10² + 6²) = √136 instead of subtracting, confusing the formula for finding a leg with finding the hypotenuse. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). When using the Pythagorean theorem, always check that you're putting the hypotenuse (the longest side) as c in the equation a² + b² = c², not one of the legs.

Question 17

In right triangle $\triangle ABC$ , $\angle C = 90^\circ$ . The legs are $AC = 9$ cm and $BC = 12$ cm, and the hypotenuse $AB = c$ is unknown. Using the given information, what is the length of side $c$ ?

$15$ cm (correct answer)
$21$ cm
$\sqrt{63}$ cm
$\sqrt{369}$ cm

Explanation: This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². This is a 3-4-5 Pythagorean triple scaled by factor 3, so we can recognize immediately that the missing side is 15 without calculation. Choice A is correct because it shows correct substitution into the Pythagorean theorem: √(9² + 12²) = √(81 + 144) = √225 = 15 cm. Choice B makes the error of adding the sides instead of adding their squares: 9 + 12 = 21 instead of √(81 + 144) = 15. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). Recognize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples to save time on calculations—if you see two sides of a triple, the third can be determined without calculation.

Question 18

A ladder is $10$ ft long and leans against a vertical wall. The bottom of the ladder is $6$ ft from the wall, forming a right triangle with the ground. What is the height (in ft) the ladder reaches on the wall?

6
8 (correct answer)
10
16

Explanation: This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Given the hypotenuse of length 10 and one leg of 6, we rearrange the Pythagorean theorem to find the other leg h: h² + 6² = 10², which gives h² + 36 = 100, so h² = 64, and taking the square root yields h = 8. Choice B is correct because it shows correct substitution into the Pythagorean theorem with the hypotenuse 10 and base 6, solving for the height of 8. Choice C incorrectly treats 10 as a leg instead of the hypotenuse, perhaps confusing the ladder's length with the height. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). When using the Pythagorean theorem, always check that you're putting the hypotenuse (the longest side) as c in the equation a² + b² = c², not one of the legs.

Question 19

A ramp rises $3$ m vertically over a horizontal run of $4$ m, forming a right triangle. Using the given information, what is the exact length (in m) of the ramp (the hypotenuse)?

5 (correct answer)
7
$\sqrt{7}$
$\sqrt{25}$

Explanation: This question tests the ability to solve right triangles using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². This is a 3-4-5 Pythagorean triple, so we can recognize immediately that the missing side is 5 without calculation. Choice A is correct because it shows correct substitution into the Pythagorean theorem with legs 3 and 4, resulting in hypotenuse √(9 + 16) = √25 = 5. Choice B calculates √(3² + 4²) as √3² + √4² = 3 + 2 = 5, but actually errs in method while getting the number right; a true distractor might add instead, like 3 + 4 = 7. Key to right triangle problems: first identify the right angle and hypotenuse (longest side, opposite the right angle), then decide whether you have enough information for Pythagorean theorem (two sides known) or need trigonometry (one side and one angle known). Recognize common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples to save time on calculations—if you see two sides of a triple, the third can be determined without calculation.

Question 20

In right triangle $\triangle ABC$ , $\angle C = 90^\circ$ . If $AC = 4$ and $BC = 4$ , what is the measure of $\angle A$ (in degrees)?

$30^\circ$
$45^\circ$ (correct answer)
$60^\circ$
$90^\circ$

Explanation: This question tests the ability to solve right triangles using both the Pythagorean theorem and trigonometric ratios. When solving right triangles, identify what is given (sides, angles) and what is unknown, then select the Pythagorean theorem if two sides are known or a trigonometric ratio if one side and one angle are known. Since AC = 4 and BC = 4, this is an isosceles right triangle, and we can find angle A using tan(A) = opposite/adjacent = BC/AC = 4/4 = 1, which means angle A = 45°. Choice B is correct because in an isosceles right triangle where the two legs are equal, both acute angles must be 45° (since they sum to 90°). Choice C incorrectly assumes this might be a 30-60-90 triangle, but in that special triangle, the sides are in ratio 1:√3:2, not 1:1:√2 as we have here. In a right triangle, the two acute angles are complementary, meaning they sum to 90°, and when the legs are equal, each acute angle must be 45°.