This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, finding the hypotenuse by plugging in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or finding a leg by rearranging to a² = c² - b², such as if c = 13 and b = 5, then a² = 169 - 25 = 144, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve 9² + h² = 15² to get h = 12 ft. In this specific problem, the right triangle has a hypotenuse of 17 units and one leg of 8 units, so the other leg is found using a² = 17² - 8² = 289 - 64 = 225, a = √225 = 15 units. The correct setup identifies c = 17 and b = 8, rearranges to a² = c² - b², calculates squares, subtracts, and takes square root to get 15 units, matching choice B. A common error might be adding squares instead of subtracting, like 289 + 64 = 353, or not squaring and doing 17 - 8 = 9, or choosing 289 without square root. To solve these problems, follow these steps: (1) identify the right triangle with 90° angle, (2) label legs and hypotenuse correctly, (3) identify knowns and unknown, (4) set up a² = c² - b², (5) calculate accordingly, (6) verify 15 is between 8 and 17. Common mistakes: forgetting squares, wrong hypotenuse, arithmetic slips, negative length.

This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, finding the hypotenuse by plugging in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or finding a leg by rearranging to a² = c² - b², such as if c = 13 and b = 5, then a² = 169 - 25 = 144, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve 9² + h² = 15² to get h = 12 ft. In this specific 3D problem, the rectangular box measures 3 in., 4 in., 12 in., so the space diagonal is d = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13 in. The correct setup extends Pythagorean to 3D by finding the diagonal of the base (√(3² + 4²) = 5) then with height √(5² + 12²) = √(25 + 144) = √169 = 13 in., or directly √(a² + b² + c²), matching choice A. A common error might be just adding dimensions 3 + 4 + 12 = 19, or using only two sides like √(3² + 4²) = 5 and stopping, or confusing with surface diagonal. To solve these problems, follow these steps: (1) identify 3D right triangle path through space, (2) label dimensions as three perpendicular legs, (3) known all three, unknown space diagonal, (4) set up d = √(a² + b² + c²), (5) calculate squares, add, square root, (6) verify 13 in. longer than 12 in. Common mistakes: not using all dimensions, arithmetic errors, forgetting it's 3D extension, negative root.

This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, finding the hypotenuse by plugging in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or finding a leg by rearranging to a² = c² - b², such as if c = 13 and b = 5, then a² = 169 - 25 = 144, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve 9² + h² = 15² to get h = 12 ft. In this specific ramp problem, the ramp is 10 ft (hypotenuse) and height is 6 ft (one leg), so the horizontal distance is b = √(10² - 6²) = √(100 - 36) = √64 = 8 ft. The correct setup identifies c = 10 ft and a = 6 ft, rearranges to b² = c² - a², calculates, and takes square root to get 8 ft, matching choice B. A common error might be adding squares instead, like 100 + 36 = 136, or not squaring and doing 10 - 6 = 4, or choosing 100 without root. To solve these problems, follow these steps: (1) identify right triangle from ramp and ground, (2) label legs as height and base, hypotenuse as ramp, (3) knowns: hypotenuse and height, unknown base, (4) set up b² = c² - a², (5) calculate, (6) verify 8 ft makes sense with 10 ft ramp. Common mistakes: no squares, wrong subtraction, arithmetic errors, negative value.

This question tests applying the Pythagorean theorem $a^2 + b^2 = c^2$ to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using $a^2 + b^2 = c^2$; for example, finding the hypotenuse by plugging in the legs like $6^2 + 8^2 = 36 + 64 = 100 = c^2$, so $c = \sqrt{100} = 10$, or finding a leg by rearranging to $a^2 = c^2 - b^2$, such as if c = 13 and b = 5, then $a^2 = 169 - 25 = 144$, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve $9^2 + h^2 = 15^2$ to get h = 12 ft. In this specific problem, the right triangle has hypotenuse 25 cm and one leg 7 cm, so the other leg is $a = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24$ cm. The correct setup identifies c = 25 cm, b = 7 cm, uses $a^2 = c^2 - b^2$, calculates, and square roots to 24 cm, matching choice B. A common error might be subtracting without squaring, like 25 - 7 = 18, or adding to 674 and choosing wrong, or picking 625 without root. To solve these problems, follow these steps: (1) identify right triangle, (2) label correctly, (3) known hypotenuse and leg, unknown leg, (4) set up $a^2 = c^2 - b^2$, (5) calculate, (6) verify 24 cm between 7 and 25. Common mistakes: no squares, wrong operation, arithmetic like $7^2 = 14$, negative length.

This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, finding the hypotenuse by plugging in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or finding a leg by rearranging to a² = c² - b², such as if c = 13 and b = 5, then a² = 169 - 25 = 144, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve 9² + h² = 15² to get h = 12 ft. In this specific problem, the right triangle has legs of 9 and 12, so the hypotenuse is c = √(9² + 12²) = √(81 + 144) = √225 = 15. The correct setup uses a = 9, b = 12 as legs, calculates c² = 81 + 144 = 225, then c = √225 = 15, matching choice A. A common error might be adding without squaring, like 9 + 12 = 21, or multiplying to 108 and choosing wrong, or forgetting square root and picking 225. To solve these problems, follow these steps: (1) identify right triangle, (2) label legs and hypotenuse, (3) known legs, unknown hypotenuse, (4) set up a² + b² = c², (5) calculate, (6) verify 15 longer than 12. Common mistakes: a + b = c, wrong squaring like 9² = 18, no root, mislabeling sides.

This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, finding the hypotenuse by plugging in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or finding a leg by rearranging to a² = c² - b², such as if c = 13 and b = 5, then a² = 169 - 25 = 144, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve 9² + h² = 15² to get h = 12 ft. In this specific problem, the right triangle has a hypotenuse of 13 m and one leg of 5 m, so we find the other leg using a² = 13² - 5², giving 169 - 25 = 144, so a = √144 = 12 m. The correct setup involves identifying c = 13 m as the hypotenuse and b = 5 m as one leg, then rearranging to a² = c² - b², calculating the squares, subtracting, and taking the square root to get 12 m, which matches choice B. A common error might be subtracting incorrectly, like 13 - 5 = 8, or forgetting to take the square root and choosing 144 m, or confusing which side is the hypotenuse and using the formula wrong. To solve these problems, follow these steps: (1) identify the right triangle with a 90° angle, (2) label the sides with legs a and b forming the right angle and hypotenuse c opposite it as the longest side, (3) identify the knowns (hypotenuse and one leg) and unknown (other leg), (4) set up a² = c² - b², (5) calculate by squaring the knowns, subtracting, and taking the square root, (6) verify it makes sense, like 12 m being between 5 m and 13 m. Common mistakes include using a + b = c without squaring, misidentifying the hypotenuse, arithmetic errors in squaring or subtracting, or taking a negative square root.

This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, finding the hypotenuse by plugging in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or finding a leg by rearranging to a² = c² - b², such as if c = 13 and b = 5, then a² = 169 - 25 = 144, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve 9² + h² = 15² to get h = 12 ft. In this specific problem, the rectangular poster is 6 inches wide and 8 inches tall, forming a right triangle with the diagonal as hypotenuse, so d² = 6² + 8² = 36 + 64 = 100, d = √100 = 10 inches. The correct setup treats the width and height as legs a = 6 in. and b = 8 in., with d as c, then squares them, adds, and takes the square root to get 10 in., matching choice B. A common error might be multiplying instead of adding squares, like 6 × 8 = 48, or forgetting the square root and choosing something like 100, or not recognizing the diagonal forms a right triangle. To solve these problems, follow these steps: (1) identify the right triangle in the rectangle's diagonal, (2) label legs as width and height, hypotenuse as diagonal, (3) identify knowns (legs) and unknown (diagonal), (4) set up a² + b² = d², (5) calculate squares, add, square root, (6) verify reasonable, like 10 in. longer than 8 in. Common mistakes: a + b = c without squares, arithmetic errors like 6² = 36 but adding wrong, confusing sides, negative root.

This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, finding the hypotenuse by plugging in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or finding a leg by rearranging to a² = c² - b², such as if c = 13 and b = 5, then a² = 169 - 25 = 144, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve 9² + h² = 15² to get h = 12 ft. In this specific problem, walking 3 blocks east and 4 blocks north forms a right triangle, so the straight-line distance is the hypotenuse d = √(3² + 4²) = √(9 + 16) = √25 = 5 blocks. The correct setup treats the directions as legs a = 3 and b = 4, with d as c, squares them, adds, and takes square root to get 5 blocks, matching choice A. A common error might be just adding 3 + 4 = 7 without squaring, or subtracting to get 1, or not recognizing the right angle turn implies a right triangle. To solve these problems, follow these steps: (1) identify the right triangle from perpendicular paths, (2) label legs as east and north distances, hypotenuse as straight line, (3) identify known legs and unknown hypotenuse, (4) set up a² + b² = c², (5) calculate squares, add, square root, (6) verify 5 is longer than 4 but shorter than path walked. Common mistakes: a + b = c, arithmetic like 3² = 6, forgetting square root, wrong setup.

This question tests applying the Pythagorean theorem $a^2 + b^2 = c^2$ to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using $a^2 + b^2 = c^2$; for example, finding the hypotenuse by plugging in the legs like $6^2 + 8^2 = 36 + 64 = 100 = c^2$, so $c = \sqrt{100} = 10$, or finding a leg by rearranging to $a^2 = c^2 - b^2$, such as if c = 13 and b = 5, then $a^2 = 169 - 25 = 144$, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve $9^2 + h^2 = 15^2$ to get h = 12 ft. In this specific problem, the right triangle has legs of 6 cm and 8 cm, and we need to find the hypotenuse using $6^2 + 8^2 = c^2$, giving 36 + 64 = 100, so $c = \sqrt{100} = 10$ cm. The correct setup involves identifying the legs as a = 6 cm and b = 8 cm, with c as the hypotenuse, then calculating the squares, adding them, and taking the square root to get 10 cm, which matches choice A. A common error might be adding without squaring, like 6 + 8 = 14, or forgetting the square root and choosing 100 cm, or confusing legs and hypotenuse by using the wrong sides in the formula. To solve these problems, follow these steps: (1) identify the right triangle with a 90° angle, (2) label the sides with legs a and b forming the right angle and hypotenuse c opposite it as the longest side, (3) identify the knowns (two legs) and unknown (hypotenuse), (4) set up $a^2 + b^2 = c^2$, (5) calculate by squaring the knowns, adding, and taking the square root, (6) verify it makes sense, like 10 cm being longer than both legs. Common mistakes include using a + b = c without squaring, misidentifying the hypotenuse, arithmetic errors in squaring or adding, or taking a negative square root.

This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, finding the hypotenuse by plugging in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or finding a leg by rearranging to a² = c² - b², such as if c = 13 and b = 5, then a² = 169 - 25 = 144, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve 9² + h² = 15² to get h = 12 ft. In this specific ladder problem, the ladder is 15 ft (hypotenuse) and the base is 9 ft from the wall (one leg), so we find the height using h² = 15² - 9², giving 225 - 81 = 144, so h = √144 = 12 ft. The correct setup involves recognizing the right triangle formed by the wall, ground, and ladder, with c = 15 ft and b = 9 ft, then calculating h² = c² - b² and taking the square root to get 12 ft, which matches choice B. A common error might be adding instead of subtracting, like 15 + 9 = 24, or not squaring the values and choosing 6 ft, or failing to identify the right triangle setup. To solve these problems, follow these steps: (1) identify the right triangle with a 90° angle implied by the wall and ground, (2) label the sides with legs as base and height, hypotenuse as ladder, (3) identify knowns (hypotenuse and base) and unknown (height), (4) set up h² = c² - b², (5) calculate by squaring, subtracting, and taking square root, (6) verify it makes sense, like 12 ft being less than the 15 ft ladder. Common mistakes include forgetting to square, wrong arithmetic like 9² = 18, confusing hypotenuse, or taking negative length.