Properties of Right Triangles - SAT Math

A right triangle has legs of lengths 3 cm and 4 cm. What is the length of the hypotenuse?
Answer: 5 cm This question asks for the hypotenuse length of a right triangle with legs of 3 cm and 4 cm. Use the Pythagorean theorem: $a^2 + b^2 = c^2$ , where c is the hypotenuse. Substituting the values: $3^2 + 4^2 = 9 + 16 = 25$ , so $c = \sqrt{25} = 5$ cm. This is a classic 3-4-5 right triangle, one of the most common Pythagorean triples. A common error is forgetting to take the square root of the final sum or mixing up which sides are legs versus hypotenuse. Recognizing common Pythagorean triples like 3-4-5 can help you solve these problems quickly on the SAT.
A square has side length $12$ . What is the length of the diagonal of the square?
Answer: $12\sqrt{2}$ This problem asks for the diagonal of a square with side length 12. A square's diagonal creates two 45°-45°-90° triangles, where the diagonal is the hypotenuse and two sides of the square are the legs. In a 45°-45°-90° triangle, the hypotenuse equals the leg times √2. Therefore, diagonal = 12√2. We can verify using the Pythagorean theorem: d² = 12² + 12² = 144 + 144 = 288, so d = √288 = √(144 × 2) = 12√2. A common error is to double the side length (getting 24) instead of multiplying by √2. Remember that a square's diagonal is always side × √2.
A right triangle has hypotenuse $13$ and one leg $5$ . What is the length of the other leg?
Answer: $12$ 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.
A ladder leans against a vertical wall. The bottom of the ladder is 6 feet from the wall, and the ladder is 10 feet long. Assuming the ground is level and the wall is perpendicular to the ground, how high up the wall does the ladder reach, in feet?
Answer: 8 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).
In the coordinate plane, a right triangle has vertices $P(0,0)$ , $Q(0,9)$ , and $R(12,0)$ . What is the length of hypotenuse $QR$ ?
Answer: $15$ We need to find the distance between points Q(0,9) and R(12,0), which forms the hypotenuse of the right triangle. Using the distance formula (which is derived from the Pythagorean theorem): d = √[(x₂-x₁)² + (y₂-y₁)²] = √[(12-0)² + (0-9)²] = √[144 + 81] = √225 = 15. The right angle is at P(0,0), with legs PQ = 9 and PR = 12. A common error is to add the coordinates directly or confuse which side is the hypotenuse. When working with coordinate geometry, always verify which vertex contains the right angle by checking that the legs are perpendicular.
A right triangle has a hypotenuse of length $13$ and one leg of length $5$ . What is the area of the triangle?
Answer: $30$ We need to find the area of a right triangle given the hypotenuse (13) and one leg (5). First, we'll use the Pythagorean theorem to find the other leg: 13² = 5² + b², so 169 = 25 + b², giving us b² = 144, and b = 12. The area of a right triangle is (1/2) × base × height = (1/2) × 5 × 12 = 30. A common mistake is trying to use the hypotenuse in the area formula, but remember that area requires the two perpendicular sides (legs). For right triangles, always find all three sides before calculating area if only two sides are given.
A right triangle is shown with a right angle at $B$ . The legs are $AB=8$ and $BC=15$ , and the hypotenuse is $AC=?$ . What is the value of $AC$ ?
Answer: $17$ 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.
In a 45-45-90 triangle, if one leg is 6 units, what is the length of the hypotenuse?
Answer: 6√2 units This question asks for the hypotenuse of a 45-45-90 triangle where one leg is 6 units. In a 45-45-90 triangle, the sides are in the ratio 1:1:√2, where the legs are equal and the hypotenuse is √2 times the length of each leg. Since one leg is 6 units, the hypotenuse = 6√2 units. You could also verify this using the Pythagorean theorem: 6² + 6² = 36 + 36 = 72, so c = √72 = √(36 × 2) = 6√2. A common error is multiplying by 2 instead of √2 or forgetting the special ratio entirely. Memorizing the 45-45-90 triangle ratio saves time compared to using the Pythagorean theorem every time.
A ramp forms a right triangle with the ground. The ramp is $13$ ft long and reaches a platform that is $5$ ft above the ground. How far is the base of the ramp from the point directly below the platform, in feet?
Answer: $12$ The question asks for the base distance of a 13 ft ramp reaching 5 ft high. This forms a right triangle with hypotenuse 13 ft and one leg 5 ft, so find the other leg using Pythagorean theorem. Base² = 13² - 5² = 169 - 25 = 144, base = √144 = 12 ft. The base is 12 feet away. Errors include adding squares instead, leading to incorrect lengths like 18. Recognize the 5-12-13 triple for faster solving in multiple-choice scenarios.
A rectangle has length $13$ and width $5$ . What is the length of the diagonal of the rectangle?
Answer: $\sqrt{194}$ This problem asks for the diagonal of a rectangle with length 13 and width 5. The diagonal of a rectangle forms the hypotenuse of a right triangle with the length and width as legs. Using the Pythagorean theorem: d² = 13² + 5² = 169 + 25 = 194, so d = √194. Since 194 = 2 × 97 and 97 is prime, √194 cannot be simplified further. A common error is to calculate √169 = 13 or √144 = 12 by mistake. When the result is an irrational number, leave it in radical form unless asked to approximate.
A right triangle has legs $3a0$ and $4a0$ . What is the length of the hypotenuse?
Answer: $5a0$ 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.
A right triangle has hypotenuse $c=26$ and one leg $a=10$ . What is the length of the other leg $b$ ?
Answer: $24$ This problem gives us a right triangle with hypotenuse c = 26 and one leg a = 10, asking for the other leg b. Using the Pythagorean theorem: $a^2 + b^2 = c^2$ , we get $10^2 + b^2 = 26^2$ , which gives us $100 + b^2 = 676$ . Solving for b²: $b^2 = 676 - 100 = 576$ , so $b = \sqrt{576} = 24$ . Note that this is a 10-24-26 triangle, which is twice the 5-12-13 Pythagorean triple. A common mistake is subtracting 26 - 10 = 16 directly. When given the hypotenuse and one leg, always subtract the squares, not the values themselves.
A ladder leans against a wall. The base of the ladder is $6$ feet from the wall, and the top of the ladder touches the wall $8$ feet above the ground. How long is the ladder, in feet?
Answer: $10$ This problem describes a ladder forming a right triangle with the wall and ground, asking for the ladder's length. The ladder is the hypotenuse, with legs of 6 feet (horizontal distance from wall) and 8 feet (vertical height on wall). Using the Pythagorean theorem: c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 feet. This is a 6-8-10 triangle, which is twice the 3-4-5 Pythagorean triple. A common error is adding 6 + 8 = 14 instead of using the Pythagorean theorem. Visualizing the problem helps identify the right triangle structure.
A right triangle has legs of lengths $9$ and $40$ . What is its area, in square units?
Answer: $180$ This problem asks for the area of a right triangle with legs of lengths 9 and 40. The area formula for a right triangle is $A = \frac{1}{2} \times \text{base} \times \text{height}$ , where the two legs serve as base and height. Therefore, $A = \frac{1}{2} \times 9 \times 40 = \frac{1}{2} \times 360 = 180$ square units. A common mistake is to first find the hypotenuse ( $41$ ) and try to use it in the area calculation, or to forget to divide by 2. For right triangles, always use the two legs for the area calculation, as they are perpendicular to each other.
A ramp rises 3 feet vertically from the ground to a platform. The ramp itself is 6 feet long. How far horizontally from the platform's base does the ramp start, in feet?
Answer: $3\sqrt{3}$ We have a right triangle where the ramp is the hypotenuse (6 feet) and the vertical rise is one leg (3 feet), and we need the horizontal distance. Using the Pythagorean theorem: 3² + h² = 6², which gives us 9 + h² = 36, so h² = 27, and h = √27 = √(9×3) = 3√3 feet. A common mistake is assuming the horizontal distance equals the vertical rise in ramp problems. Always identify which measurement represents the hypotenuse (usually the ramp length itself) before applying the theorem.
A right triangle has legs of lengths $9$ inches and $12$ inches. What is the length of the hypotenuse, in inches?
Answer: $15$ This question asks for the hypotenuse of a right triangle given the two legs. We use the Pythagorean theorem: a² + b² = c², where a and b are legs and c is the hypotenuse. Substituting the given values: 9² + 12² = 81 + 144 = 225, so c = √225 = 15. A common error is adding the legs directly (9 + 12 = 21) instead of using the Pythagorean theorem. Remember that 9-12-15 is a multiple of the 3-4-5 Pythagorean triple, which can help verify your answer quickly.
A right triangle has hypotenuse $10$ and one acute angle of $45^\circ$ . What is the area of the triangle?
Answer: $25$ This problem involves a 45-45-90 special right triangle with hypotenuse 10. In a 45-45-90 triangle, the legs are equal and each leg equals the hypotenuse divided by √2, so each leg = 10/√2 = 10√2/2 = 5√2. The area formula for a triangle is A = (1/2) × base × height = (1/2) × 5√2 × 5√2 = (1/2) × 25 × 2 = 25. Students often forget that in a 45-45-90 triangle, both legs are equal, or they may incorrectly apply the special triangle ratios. Remember that the area calculation requires multiplying the two legs, not using the hypotenuse directly.
A 45-45-90 right triangle has hypotenuse length $10\sqrt{2}$ . What is the length of each leg?
Answer: $10$ 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.
On a coordinate plane, point A is at (0, 0) and point B is at (3, 4). What is the distance between these two points?
Answer: 5 units This question asks for the distance between points A(0, 0) and B(3, 4) on a coordinate plane. Use the distance formula, which is derived from the Pythagorean theorem: d = √[(x₂-x₁)² + (y₂-y₁)²]. Substituting the coordinates: d = √[(3-0)² + (4-0)²] = √[3² + 4²] = √[9 + 16] = √25 = 5 units. This creates a right triangle with legs of length 3 and 4, giving us the familiar 3-4-5 triangle. A common error is forgetting to take the square root or mixing up the coordinate differences. The distance formula is essentially the Pythagorean theorem applied to coordinate geometry.
Points $A(2,-1)$ and $B(8,7)$ are endpoints of a segment. What is the distance from $A$ to $B$ ?
Answer: $10$ To find the distance between two points, we use the distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]. Substituting A(2,-1) and B(8,7): d = √[(8-2)² + (7-(-1))²] = √[6² + 8²] = √[36 + 64] = √100 = 10. A common error is forgetting to handle the negative y-coordinate correctly: 7-(-1) = 8, not 6. The distance formula is derived from the Pythagorean theorem, treating the coordinate differences as legs of a right triangle.

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