Find Exact Values of p, q, r in Special Right Triangles (45-45-90 and 30-60-90)

Question

Find the exact value of each part labeled with a variable (p, q, r) in the figure. The figure contains two right triangles: one is a 45-45-90 right triangle with one leg length 33, and the other is a 30-60-90 right triangle sharing the hypotenuse of the first triangle as one of its legs. Simplify answers, including any radicals. Use integers or fractions for any numbers in the expressions.

Accepted Answer

p=33, q=33√3, r=33√2

Answer

Step 1: Solve for p and r using the left 45-45-90 right triangle. In a 45-45-90 triangle, the two legs are equal, so since one leg is 33, p = 33. Step 2: Calculate r (the hypotenuse of the 45-45-90 triangle). The hypotenuse of a 45-45-90 triangle is leg length × √2, so r = 33×√2 = 33√2. Step 3: Solve for q using the right 30-60-90 right triangle. Here, r is the shorter leg (opposite the 30° angle). In a 30-60-90 triangle, the longer leg is shorter leg × √3, so q = 33√2 × √3 = 33√6.