Question

what is the length of \\(\overline{de}\\) ? there are two triangles. the upper triangle is \\(\triangle fde\\) with \\(\angle f = 44^\circ\\), \\(fd = 8\\), \\(fe = 8\\), and \\(de = x\\). the lower triangle is \\(\triangle zxy\\) with \\(\angle z = 44^\circ\\), \\(zx = 12\\), \\(zy = 12\\), and \\(xy = 9\\). the options are a. 7, b. 6, c. 8, d. 5

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle FDE \) and \( \triangle ZXY \) have two sides equal (\( FD = FE = 8 \), \( ZX = ZY = 12 \)) and the included angle \( \angle F = \angle Z = 44^\circ \), so by SAS similarity, they are similar.

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So \( \frac{DE}{XY}=\frac{FD}{ZX} \). Substitute \( XY = 9 \), \( FD = 8 \), \( ZX = 12 \): \( \frac{x}{9}=\frac{8}{12} \).

Step3: Solve for \( x \)

Simplify \( \frac{8}{12}=\frac{2}{3} \), then \( x = 9\times\frac{2}{3}=6 \).

Answer:

B. 6