Question

what is the length of \\(overline{bc}\\)? \\(\bigcirc\\) 9 units \\(\bigcirc\\) 11 units \\(\bigcirc\\) 15 units \\(\bigcirc\\) 16 units

Explanation:

Step1: Identify right triangles

We have a right triangle with hypotenuse \( AB = 17 \) and one leg \( AC = 8 \). Let the other leg (height to hypotenuse) be \( x \), but we need \( BC \). Wait, actually, there's a right triangle \( ABC \)? Wait, no, the diagram shows a right triangle with hypotenuse \( AB = 17 \), leg \( AC = 8 \), and we need \( BC \). Wait, maybe it's a right triangle, so use Pythagorean theorem: \( AB^2 = AC^2 + BC^2 \)? Wait, no, maybe the right angle is at \( C \)? Wait, the diagram has a right angle, so let's assume triangle \( ABC \) is right-angled, with \( AC = 8 \), \( AB = 17 \), and we need \( BC \).

Step2: Apply Pythagorean theorem

The Pythagorean theorem states that in a right triangle, \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. Let \( AC = 8 \), \( AB = 17 \), so \( BC^2 + AC^2 = AB^2 \). Wait, no, if \( AB \) is the hypotenuse, then \( BC^2 + AC^2 = AB^2 \)? Wait, no, maybe \( AC \) is one leg, \( BC \) is another leg, and \( AB \) is hypotenuse. So \( BC = \sqrt{AB^2 - AC^2} \).

Calculating: \( AB = 17 \), \( AC = 8 \), so \( AB^2 = 17^2 = 289 \), \( AC^2 = 8^2 = 64 \). Then \( BC^2 = 289 - 64 = 225 \), so \( BC = \sqrt{225} = 15 \).

Answer:

15 units