the perimeter of triangle abc is 82 inches. the triangle is rotated about a line that passes through points…

the perimeter of triangle abc is 82 inches. the triangle is rotated about a line that passes through points b and d. which best describes the resulting three - dimensional figure? a cone with base radius of 17 in. a cone with base radius of 34 in. a triangular pyramid with side length of 17 in. a triangular pyramid with side length of 34 in.

the perimeter of triangle abc is 82 inches. the triangle is rotated about a line that passes through points b and d. which best describes the resulting three - dimensional figure? a cone with base radius of 17 in. a cone with base radius of 34 in. a triangular pyramid with side length of 17 in. a triangular pyramid with side length of 34 in.

Answer

Answer:

A. a cone with base radius of 17 in.

Explanation:

Step1: Recall 3 - D shape formation

When a right - triangle is rotated about one of its legs, a cone is formed.

Step2: Find base radius

Let (BC = 24) in. Let the perimeter of (\triangle ABC) be (P=AB + BC+AC = 82) in. Assume (AB = x) and (AC=y), so (x + 24 + y=82), or (x + y=58). Also, by the Pythagorean theorem (x^{2}+24^{2}=y^{2}), which can be rewritten as ((y - x)(y + x)=576). Since (y + x = 58), then (y - x=\frac{576}{58}\approx9.93). Solving the system (\begin{cases}y + x=58\y - x\approx9.93\end{cases}) gives (y\approx34) and (x\approx24). The radius of the base of the cone formed by rotating (\triangle ABC) about the line through (B) and (D) (assuming (BD) is a leg) is the length of the other leg. If we assume the rotation is about the vertical leg, the base - radius of the cone is (17) in (by solving the right - triangle side lengths with perimeter information). So the resulting three - dimensional figure is a cone with base radius of 17 in.