Short Answer 
Similar right triangles have equal ratios of corresponding side lengths, allowing proportions to find unknown sides. The Pythagorean theorem helps calculate side lengths in right triangles through (a^2 + b^2 = h^2), and the geometric mean (h = sqrt{ab}) aids in determining lengths in altitude-related scenarios within these triangles.
Step 1: Understand Similar Right Triangles
Similar right triangles are triangles that have the same shape but not necessarily the same size. In these triangles, the ratios of corresponding side lengths are equal. This means that if you have two similar right triangles, you can set up proportions to find unknown side lengths. For instance, if triangle A has sides (a) and (b), and triangle B has sides (ka) and (kb) (where (k) is a scale factor), the proportions can help you understand their measurements.
Step 2: Apply the Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the hypotenuse. This relationship is crucial when calculating unknown side lengths. Specifically, given sides (a) and (b), the hypotenuse (h) can be found using the formula: (a^2 + b^2 = h^2). This equation helps to establish the lengths of all sides in the triangle and can be used to confirm the triangle’s right-angle nature.
Step 3: Utilize the Geometric Mean
The Geometric Mean is an essential concept when analyzing the lengths in similar triangles. It can give you the length of segments in right triangles, especially in altitude-related scenarios. For any right triangle, you can find the segments created by an altitude that divides the triangle into two smaller triangles. The length of the altitude is given by the formula: (h = sqrt{ab}), where (a) and (b) are the lengths of the two segments formed by the altitude. This relationship aids in finding unknown heights or lengths efficiently.