Question

12)
(figure of a geometric diagram with triangle and right angles, labeled with lengths: 10 in., 7 in., 8 in., and x for the base of the larger right - angled figure)

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle CBD \) and \( \triangle CAE \) are similar (by AA similarity, as \( \angle CBD = \angle CAE = 90^\circ \) and \( \angle C \) is common).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \( \frac{CD}{CE} = \frac{BD}{AE} \). First, find \( CE = CD + DE = 10 + 8 = 18 \) in. Then, substitute \( CD = 10 \), \( CE = 18 \), \( BD = 7 \), and \( AE = x \) into the proportion: \( \frac{10}{18} = \frac{7}{x} \).

Step3: Solve for \( x \)

Cross - multiply: \( 10x = 18\times7 \). Calculate \( 18\times7 = 126 \), so \( 10x = 126 \). Then, \( x=\frac{126}{10}=12.6 \).

Answer:

\( 12.6 \) inches