Final answer:
To find the length of the shortest leg in a right triangle with the given hypotenuse, the Pythagorean theorem is applied. The angle of depression is determined using the tangent function, which compares the lengths of opposite and adjacent sides of the triangle.
Step-by-step explanation:
The student's question involves using the Pythagorean theorem to solve for the length of the shortest leg of a right triangle, given that the shortest leg is two-thirds the size of the hypotenuse, and to find the angle of depression using trigonometric functions. Let's denote the hypotenuse as 'c' and the shortest leg as 'a', given that 'a' is two-thirds of 'c', we can write 'a=(2/3)c'. Applying the Pythagorean theorem 'a² + b² = c²', and solving for 'a', we find the length of the shortest leg.
To find the angle of depression, we can use the trigonometric function tangent (tan) because we have the lengths of the opposite and adjacent sides of a right triangle. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Thus, tan(angle) = opposite/adjacent. To find the angle itself, we use the inverse tangent function (tan⁻¹ or arctan). The angle of depression, rounded to the nearest degree, corresponds to the angle of elevation from Lisa's line of sight to the hoop.