SAT Math Quiz: Properties Of Right Triangles

What this quiz covers

This quiz focuses on Properties Of Right Triangles, giving you a quick way to practice the rules, question types, and explanations that matter most for SAT Math.

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.

Question 1

1. $5\sqrt{2}$
2. $10\sqrt{2}$
3. $5$
4. $10$ (correct answer)

Explanation: The question asks for the length of each leg in a 45-45-90 right triangle with hypotenuse 10√2. In a 45-45-90 triangle, the legs are equal and each is hypotenuse / √2. Divide: leg = 10√2 / √2 = 10. Both legs are 10 units long. Errors often occur by confusing ratios with 30-60-90 triangles, leading to options like 5√2. Remember the special ratio for 45-45-90 is 1:1:√2 to select the correct choice efficiently.

Question 2

1. $17$ (correct answer)
2. $23$
3. 289
4. 169

Explanation: This problem shows a right triangle with right angle at B, legs AB = 8 and BC = 15, asking for hypotenuse AC. Using the Pythagorean theorem: AC² = AB² + BC² = 8² + 15² = 64 + 225 = 289. Therefore, AC = √289 = 17. This is the 8-15-17 Pythagorean triple. Common errors include adding the legs (8 + 15 = 23) instead of using the Pythagorean theorem. Recognizing common Pythagorean triples like 8-15-17 can save time on tests.

Question 3

1. $18$
2. $8$
3. $12$ (correct answer)
4. $\sqrt{144}$

Explanation: We need to find the missing leg when given the hypotenuse (13) and one leg (5). Using the Pythagorean theorem: $a^2 + b^2 = c^2$, we rearrange to find $b^2 = c^2 - a^2 = 13^2 - 5^2 = 169 - 25 = 144$, so $b = \sqrt{144} = 12$. This is a 5-12-13 Pythagorean triple, which appears frequently on tests. A common error is subtracting directly ($13 - 5 = 8$) instead of using squares. Always remember that the hypotenuse is the longest side in a right triangle.

Question 4

1. $15$
2. $30$ (correct answer)
3. $60$
4. $\frac{65}{2}$

Explanation: Given a right triangle with hypotenuse 13 and one leg 5, we first find the other leg using the Pythagorean theorem. Setting up: 5² + b² = 13², we get 25 + b² = 169, so b² = 144, and b = 12. This is the classic 5-12-13 Pythagorean triple. The area of a right triangle equals (1/2) × base × height = (1/2) × 5 × 12 = 30. A common error is trying to use the hypotenuse in the area formula; remember that for right triangles, the area uses the two legs as base and height.

Question 5

1. 4
2. 8 (correct answer)
3. 16
4. $\sqrt{136}$

Explanation: This problem asks for the height the ladder reaches on the wall, forming a right triangle with the wall, ground, and ladder. We'll use the Pythagorean theorem where the ladder is the hypotenuse (10 feet), the ground distance is one leg (6 feet), and the wall height is the unknown leg. Applying the formula: $6^2 + h^2 = 10^2$, which gives us $36 + h^2 = 100$, so $h^2 = 64$, and $h = 8$ feet. A common error is confusing which measurement is the hypotenuse versus the legs. When visualizing ladder problems, remember the ladder itself is always the longest side (hypotenuse).

Question 6

1. $20$
2. $2\sqrt{100}$
3. $5\sqrt{2}$
4. $10\sqrt{2}$ (correct answer)

Explanation: By the Pythagorean theorem the hypotenuse c satisfies c^2 = 10^2 + 10^2 = 100 + 100 = 200, so c = sqrt(200). Simplify by pulling out the perfect square: sqrt(200) = sqrt(100 * 2) = 10 * sqrt(2), which is about 14.1 cm. The plain value 20 comes from adding the two legs instead of adding their squares. The expression 2 * sqrt(100) also works out to 20; it comes from splitting 200 into 2 * 100 but then taking the square root of only the 100 while leaving the 2 outside, when a radical requires the root of each factor, giving sqrt(2) * sqrt(100). And 5 * sqrt(2) is about 7.07, which is shorter than either leg and so cannot be a hypotenuse.

Question 7

1. $10$ (correct answer)
2. $\sqrt{28}$
3. $\sqrt{100}$
4. $\sqrt{52}$

Explanation: This problem asks for the distance between two points using the distance formula, which is derived from the Pythagorean theorem. The distance formula is d = √[(x₂-x₁)² + (y₂-y₁)²]. Substituting: d = √[(4-(-2))² + (-5-3)²] = √[6² + (-8)²] = √[36 + 64] = √100 = 10. Common errors include sign mistakes when subtracting coordinates or forgetting to square the differences. The distance √100 simplifies to 10, not left as √100.

Question 8

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

Explanation: A fundamental identity of trigonometry for right triangles is that the sine of an angle is equal to the cosine of its complement. $\sin(x) = \cos(90 - x)$. Therefore, if $\sin(x) = \frac{4}{5}$, then $\cos(90 - x) = \frac{4}{5}$.

Question 9

1. 6√3 units
2. 12 units (correct answer)
3. 12√3 units
4. 6 units

Explanation: This question requires knowledge of the special 30-60-90 triangle ratio relationships. In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2. Since the shorter leg measures 6 units, we multiply the ratio by 6, giving us sides of 6 : 6√3 : 12. Therefore, the hypotenuse is 12 units.

Question 10

1. 18 cm
2. 10 cm
3. 12 cm (correct answer)
4. 8 cm

Explanation: This question asks for the unknown leg of a right triangle given the hypotenuse (13 cm) and one leg (5 cm). Use the Pythagorean theorem: a² + b² = c², rearranged to solve for the missing leg: b² = c² - a². Substituting: b² = 13² - 5² = 169 - 25 = 144, so b = √144 = 12 cm. This is the famous 5-12-13 Pythagorean triple. A common error is subtracting incorrectly or forgetting to take the square root of the final result. When given the hypotenuse and one leg, always subtract the leg squared from the hypotenuse squared to find the other leg.

Question 11

1. $8$
2. $4$ (correct answer)
3. $\sqrt{34}$
4. $2$

Explanation: This ramp problem forms a right triangle where the hypotenuse is the ramp (5 ft) and one leg is the vertical rise (3 ft). We need the horizontal distance, which is the other leg. Using the Pythagorean theorem: 3² + horizontal² = 5², so horizontal² = 25 - 9 = 16, giving horizontal = 4 ft. This is a 3-4-5 right triangle. When dealing with ramps or ladders, sketch the right triangle to identify which measurement corresponds to which side.

Question 12

1. $\sqrt{776}$
2. $16$
3. $24$ (correct answer)
4. $\sqrt{576}$

Explanation: We need to find the missing leg when given the hypotenuse (26) and one leg (10). Using the Pythagorean theorem rearranged: leg² = hypotenuse² - other leg² = 26² - 10² = 676 - 100 = 576, so the missing leg = √576 = 24. A common mistake is subtracting before squaring, getting 16² = 256 instead of the correct answer. When finding a leg rather than the hypotenuse, remember to subtract the squares, not add them.

Question 13

1. 492
2. $24.5$
3. $49$ (correct answer)
4. $98$

Explanation: In a 45°-45°-90° triangle with hypotenuse 14, each leg has length 14/√2 = 14/√2 × √2/√2 = 14√2/2 = 7√2. The area = (1/2) × leg × leg = (1/2) × 7√2 × 7√2 = (1/2) × 49 × 2 = 49. A common error is forgetting that both legs are equal in an isosceles right triangle. Another approach: if hypotenuse = 14, then leg² + leg² = 196, so 2(leg²) = 196, giving leg² = 98 and area = (1/2) × 98 = 49.

Question 14

1. $2$
2. $3$
3. $6$ (correct answer)
4. $12$

Explanation: We have a right triangle with legs 2x and 3x and hypotenuse 6√13. Applying the Pythagorean theorem: (2x)² + (3x)² = (6√13)², which gives us 4x² + 9x² = 36(13) = 468. Combining like terms: 13x² = 468, so x² = 36, and therefore x = 6. Students often make errors when squaring expressions with radicals or forget to square the coefficient along with the variable. To verify your answer, substitute back: legs are 12 and 18, and 12² + 18² = 144 + 324 = 468 = (6√13)².

Question 15

1. 510 (correct answer)
2. 710
3. $50$
4. 130

Explanation: We need to find the hypotenuse of a right triangle with legs 3√10 and 4√10. Using the Pythagorean theorem: c² = a² + b², we substitute c² = (3√10)² + (4√10)² = 9(10) + 16(10) = 90 + 160 = 250. Therefore, c = √250 = √(25 × 10) = 5√10. A common mistake is to incorrectly simplify the radicals or forget to square the coefficients along with the radicals. When working with radical expressions in the Pythagorean theorem, square both the coefficient and the radical separately before adding.

Question 16

1. $14$
2. 72 (correct answer)
3. 49
4. 74

Explanation: This question involves a 45°-45°-90° special right triangle where both legs are equal. In this special triangle, if each leg has length a, the hypotenuse has length a√2. Since each leg is 7, the hypotenuse is 7√2. Students often mistakenly double the leg (getting 14) or add the legs (getting 7√4). Remember the 45-45-90 ratio: if legs are x, then hypotenuse is x√2.

Question 17

1. $16$
2. $\sqrt{136}$ (correct answer)
3. $8$
4. $\sqrt{64}$

Explanation: This question asks for the diagonal of a rectangle, which forms the hypotenuse of a right triangle with the length and width as legs. Using the Pythagorean theorem: diagonal² = 10² + 6² = 100 + 36 = 136, so diagonal = √136. Students often make arithmetic errors (like 10 + 6 = 16) or try to simplify √136 incorrectly (√136 ≠ √64 = 8). Leave the answer as √136 unless asked to approximate, as this is the exact value.

Question 18

1. $9$
2. $9\sqrt{3}$ (correct answer)
3. $18\sqrt{3}$
4. $6\sqrt{3}$

Explanation: This problem asks for the longer leg in a 30°-60°-90° triangle with hypotenuse 18. In a 30°-60°-90° triangle, the sides are in ratio 1 : √3 : 2, where the hypotenuse is 2x, the shorter leg is x, and the longer leg is x√3. Since hypotenuse = 18 = 2x, we get x = 9, so the longer leg = 9√3. Common errors include using 9 as the answer (the shorter leg) or multiplying incorrectly to get 18√3. Memorizing the 30°-60°-90° ratio pattern helps solve these problems quickly.

Question 19

1. $34$
2. $60$ (correct answer)
3. $58$
4. $26$

Explanation: To find the perimeter, we need all three sides: the two legs (10 and 24) plus the hypotenuse. Using the Pythagorean theorem: 10² + 24² = 100 + 576 = 676, so the hypotenuse = √676 = 26. The perimeter is 10 + 24 + 26 = 60. A common mistake is forgetting to find the hypotenuse first, or calculating area instead of perimeter. This uses the 5-12-13 Pythagorean triple scaled by 2.

Question 20

1. 25 m (correct answer)
2. 30 m
3. 22.5 m
4. 35 m

Explanation: This question asks for the hypotenuse of a right triangle with legs of 15 m and 20 m. Use the Pythagorean theorem: a² + b² = c², where the legs are 15 m and 20 m. Substituting: 15² + 20² = 225 + 400 = 625, so c = √625 = 25 m. This is the 3-4-5 triangle scaled by a factor of 5 (since 15 = 3×5, 20 = 4×5, and 25 = 5×5). A common error is making arithmetic mistakes with larger numbers or not recognizing the scaled Pythagorean triple. Look for patterns and multiples of common triples like 3-4-5 to check your work quickly.