We're given a right triangle with legs 4x and 3x and hypotenuse 25, and need to find x. Using the Pythagorean theorem: (4x)² + (3x)² = 25². This gives us 16x² + 9x² = 625, so 25x² = 625, and x² = 25, therefore x = 5. We can verify: legs are 4(5) = 20 and 3(5) = 15, and indeed 20² + 15² = 400 + 225 = 625 = 25². This is a scaled version of the 3-4-5 triple. Watch for problems that use variables with Pythagorean triples.

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 forgetting to take the square root of the sum, which would give 225 instead of 15. When you see a 9-12 right triangle, recognize it as a multiple of the 3-4-5 Pythagorean triple (multiply each by 3).

The question asks for the length of the hypotenuse in a 30°-60°-90° triangle where the longer leg is 12√3. Use the special ratios for a 30°-60°-90° triangle, where the sides are in the ratio x : x√3 : 2x, with the longer leg opposite the 60° angle being x√3. Here, x√3 = 12√3, so divide both sides by √3 to find x = 12. The hypotenuse is then 2x = 2 × 12 = 24. A key error is confusing the longer leg with the hypotenuse, leading to multiplying by √3 instead of finding 2x. For special triangles, memorize the ratios to quickly identify the hypotenuse as twice the shorter leg.

The question asks for the length of a ramp that rises 3 feet vertically over 10 feet horizontally. Apply the Pythagorean theorem to find the hypotenuse of this right triangle. The length l = $\sqrt{3^2 + 10^2}$ = $\sqrt{9 + 100}$ = $\sqrt{109}$ feet. This matches choice A. Errors often occur by adding instead of squaring, giving 13, or swapping legs and hypotenuse. Visualize the ramp as the hypotenuse to confirm the calculation setup.

The question asks for the height the 13-foot ladder reaches up the wall when its base is 5 feet from the wall. Use the Pythagorean theorem, treating the ladder as the hypotenuse and solving for the missing leg. The height h satisfies h² = 13² - 5² = 169 - 25 = 144, so h = √144 = 12 feet. This matches choice B. A key error is subtracting incorrectly or forgetting to take the square root, leading to 144 or adding to get 18. In ladder problems, ensure the hypotenuse is the longest side to avoid confusion.

The question asks for the perimeter of a right triangle with legs 11 and 4. First, find the hypotenuse using the Pythagorean theorem: √(11² + 4²) = √(121 + 16) = √137. Then, perimeter = 11 + 4 + √137 = 15 + √137. This matches choice A. Common errors include adding legs without the hypotenuse or squaring the sum. Calculate the hypotenuse separately before adding for accuracy.

The question asks for the length of the hypotenuse in a right triangle with legs $\sqrt{20}$ and $\sqrt{45}$, in simplest radical form. Apply the Pythagorean theorem: $$c = \sqrt{(\sqrt{20})^2 + (\sqrt{45})^2} = \sqrt{20 + 45} = \sqrt{65}.$$ Note that $\sqrt{65}$ cannot be simplified further as 65 has no perfect square factors other than 1. Simplifying the legs first to $2\sqrt{5}$ and $3\sqrt{5}$ shows $$c = \sqrt{4 \times 5 + 9 \times 5} = \sqrt{65},$$ confirming. A common mistake is adding the radicals directly without squaring. For radicals, square them early to combine under one root for simplicity.

The question asks for the length of the hypotenuse in a right triangle with legs of lengths 2 and 3. Apply the Pythagorean theorem: the hypotenuse c is given by c = √(a² + b²), where a=2 and b=3. Substitute the values: c = √(2² + 3²) = √(4 + 9) = √13. The result √13 is already in simplest radical form, as 13 has no perfect square factors other than 1. A frequent mistake is adding the legs instead of their squares, like √(2 + 3) = √5, which underestimates the hypotenuse. For quick verification on tests, remember that hypotenuse is always longer than each leg, so √13 (about 3.6) fits between 3 and the expected larger value.

This involves a 30°-60°-90° special right triangle with specific side ratios. In this triangle type, if the side opposite 30° = x, then the hypotenuse = 2x, and the side opposite 60° = x√3. Since the side opposite 30° is 5 meters, the hypotenuse = 2 × 5 = 10 meters. A common error is confusing which side corresponds to which angle or mixing up the ratios with those of a 45-45-90 triangle. Always remember: in a 30-60-90 triangle, the hypotenuse is exactly twice the shortest side.

To find the distance between two points, we use the distance formula, which is derived from the Pythagorean theorem: d = √[(x₂-x₁)² + (y₂-y₁)²]. With points (-2, 1) and (4, 9): d = √[(4-(-2))² + (9-1)²] = √[6² + 8²] = √[36 + 64] = √100 = 10. This forms a right triangle with legs of length 6 and 8, another 3-4-5 triple scaled by 2. A common error is forgetting to square the differences before adding them. The distance formula is essentially the Pythagorean theorem applied to coordinate geometry.