Question

the 2 triangles shown below are similar. the measures of ∠b and ∠y are equal, as are the measures of ∠c and ∠z. what is the length, in feet, of \\(\overline{ab}\\)? (there are two right triangles: left triangle has right angle at a, hypotenuse bc labeled 20 ft; right triangle has right angle at x, legs zx=8 ft, xy=6 ft, hypotenuse zy=10 ft)

Explanation:

Step1: Identify Similar Triangles

Triangles \(ABC\) and \(ZYX\) are similar (right triangles with two equal angles, so AA similarity). Corresponding sides are proportional.

Step2: Find Scale Factor

Hypotenuse of \(ZYX\) is \(10\) ft, hypotenuse of \(ABC\) is \(20\) ft. Scale factor \(k=\frac{20}{10}=2\).

Step3: Calculate \(AB\)

In \(ZYX\), side \(XY = 6\) ft. \(AB\) corresponds to \(XY\), so \(AB=6\times2 = 12\) ft.

Answer:

12