Question

1. in the figure shown, δabc is a right triangle. \\(\overline{bd}\\) is perpendicular to the hypotenuse \\(\overline{ac}\\). what is the length of \\(\overline{bd}\\) to the nearest tenth? figure of right triangle abc with right angle at b, ab = 20, bc = 48, and bd perpendicular to ac at d \\(\bigcirc\\) 12 \\(\bigcirc\\) 6.13 \\(\bigcirc\\) 9.25 \\(\bigcirc\\) 15.6 \\(\bigcirc\\) 21.2 \\(\bigcirc\\) 37 \\(\bigcirc\\) 18.5

Explanation:

Step1: Find the length of hypotenuse \( AC \)

In right triangle \( ABC \), by the Pythagorean theorem, \( AC = \sqrt{AB^2 + BC^2} \). Given \( AB = 20 \) and \( BC = 48 \), so \( AC = \sqrt{20^2 + 48^2} = \sqrt{400 + 2304} = \sqrt{2704} = 52 \).

Step2: Use the area formula to find \( BD \)

The area of \( \triangle ABC \) can be calculated in two ways: \( \frac{1}{2} \times AB \times BC \) and \( \frac{1}{2} \times AC \times BD \).
Set them equal: \( \frac{1}{2} \times 20 \times 48 = \frac{1}{2} \times 52 \times BD \).
Simplify: \( 20 \times 48 = 52 \times BD \), so \( BD = \frac{20 \times 48}{52} = \frac{960}{52} \approx 18.5 \).

Answer:

18.5