Final answer:
The ratios of the side lengths of right triangles change with the measure of the acute angle, as described by the Pythagorean theorem and trigonometric ratios like sine, cosine, and tangent.
Step-by-step explanation:
The student's question involves observing the ratios of the side lengths of right triangles and how these ratios change as the measure of the acute angle a changes. In the context of right triangles, the properties of the side lengths are often examined using the Pythagorean theorem and trigonometric ratios.
The Pythagorean theorem states that a² + b² = c², where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse. Trigonometry, on the other hand, explores the relationship between the angles and sides of right angled triangles. The primary trigonometric ratios are sine, cosine, and tangent, which correspond to the ratios of various sides of the triangle relative to its angles.
As the measure of the acute angle a changes, the values of these trigonometric ratios will also change, demonstrating how the side lengths of the right triangle are in a constant but variable relationship to each other, dependent on the angle measures. Specifically, for a given right triangle with angle a, the lengths of the sides opposite and adjacent to a compared to the hypotenuse will change as the angle a changes, reflecting the sine and cosine ratios respectively, while the tangent ratio will change as the ratio of the opposite side to the adjacent side.