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.

The question asks for the length of the diagonal in a square with side length 10, where the diagonal forms the hypotenuse of a right triangle with the sides as legs. Apply the Pythagorean theorem: diagonal d = √(10² + 10²) = √(100 + 100). Compute 100 + 100 = 200, so d = √200 = √(100 × 2) = 10√2. A key error is treating it as a 45-45-90 triangle but forgetting to multiply by √2, resulting in just 10. Some might double the side incorrectly to get 20. Remember that in squares, the diagonal is always side × √2, which simplifies calculations without full recomputation.

The question asks for the length of the hypotenuse AB in right triangle ABC with a right angle at C and legs AC = 9 cm and BC = 12 cm. Use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. Apply the formula: AB = √(AC² + BC²) = √(9² + 12²). Calculate 9² = 81 and 12² = 144, so 81 + 144 = 225, and √225 = 15 cm. A common error is adding the legs instead of their squares, which would give an incorrect 21 cm. Another mistake might be squaring the sum of the legs, leading to √(21)² = 21, but that's not the theorem. When facing Pythagorean problems, verify if it's a multiple of a known triple like 3-4-5, here scaled by 3 to 9-12-15.

The question asks how high up the wall a 13-foot ladder reaches when its base is 5 feet from the wall, forming a right triangle. Use the Pythagorean theorem, where the ladder is the hypotenuse, the base is one leg, and the height is the other leg. Solve for height h: h = √(13² - 5²) = √(169 - 25). Calculate 169 - 25 = 144, and √144 = 12 feet. Students might err by subtracting squares incorrectly or forgetting the square root, yielding 144 instead of 12. Another common mistake is adding the squares, giving √(169 + 25) = √194, which is irrelevant here. A useful strategy is to recognize 5-12-13 as a Pythagorean triple for quick verification.

The question asks for the length of the other leg in a right triangle with hypotenuse 14 and one leg 7. Apply the Pythagorean theorem rearranged to find the missing leg: leg2 = √(hypotenuse² - leg1²) = √(14² - 7²). Calculate 14² = 196 and 7² = 49, so 196 - 49 = 147, and √147 = √(49 × 3) = 7√3. Ensure subtraction is in the correct order, as reversing would give a negative under the square root. A frequent error is adding instead of subtracting, leading to √(196 + 49) = √245, which is incorrect. Some might divide the hypotenuse by 2, but that's not applicable here. When verifying, check if the sides satisfy a² + b² = c² after solving.

The question asks for the horizontal distance from the base of a 3-meter-high platform to the bottom of a 5-meter-long ramp, assuming level ground. Model this as a right triangle with hypotenuse 5 (ramp), one leg 3 (height), and the other leg as the base b. Use Pythagorean theorem: b = √(5² - 3²) = √(25 - 9). Calculate 25 - 9 = 16, so √16 = 4 meters. A common error is using addition instead, yielding √(25 + 9) = √34, which is incorrect. Another mistake might be treating the height as the hypotenuse, leading to invalid negatives. Recognize 3-4-5 as a Pythagorean triple for efficient solving without recalculation.

This problem asks how high the ladder reaches on the wall, which is one leg of a right triangle. The ladder forms the hypotenuse (20 feet), and the distance from wall to ladder base is the other leg (12 feet). Using the Pythagorean theorem: 12² + h² = 20², so 144 + h² = 400, giving h² = 256 and h = 16 feet. A common mistake is subtracting the base distance from the ladder length (20 - 12 = 8) instead of using the Pythagorean theorem. Visualizing the right triangle formed by the wall, ground, and ladder helps identify which measurements correspond to which sides.

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.

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.

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.