Question

9.
(image of two right triangles sharing the same right angle vertex r, with the larger triangle having height 12, base 9, hypotenuse 15, and the smaller triangle having base 6, hypotenuse 10, and unknown height at b. handwritten 10.5 at the bottom.)

Explanation:

Step1: Identify Similar Triangles

Triangles \( SRT \) and \( BRC \) are similar (by AA similarity, as they share angle \( R \) and have parallel sides \( SB \) and \( RC \) with \( ST \) and \( BC \), so corresponding angles are equal).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. Let the length of \( BR \) be \( x \). Then \( SR = 12 \), \( ST = 15 \), \( BC = 10 \), \( RT = 9 \), \( RC = 6 \). The proportion is \( \frac{BR}{SR}=\frac{RC}{RT}=\frac{BC}{ST} \). Using \( \frac{x}{12}=\frac{6}{9} \) (or \( \frac{x}{12}=\frac{10}{15} \), both simplify to the same).

Step3: Solve for \( x \)

Simplify \( \frac{6}{9}=\frac{2}{3} \), so \( \frac{x}{12}=\frac{2}{3} \). Cross - multiply: \( 3x = 12\times2 = 24 \), then \( x=\frac{24}{3}=8 \). Wait, but the handwritten answer is 10.5? Wait, maybe I misidentified the sides. Wait, \( SR = 12 \), \( RT = 9 \), \( RC = 6 \), \( BC = 10 \), \( ST = 15 \). Let's use \( \frac{BR}{SR - BR}=\frac{RC}{RT - RC}=\frac{BC}{ST - BC} \)? No, better: Let \( BR = x \), so \( SB=12 - x \). The triangles \( SBC \) and \( SRT \)? Wait, no, \( B \) is on \( SR \), \( C \) is on \( RT \), so \( BC \parallel ST \), so \( \triangle BRC \sim \triangle SRT \). So \( \frac{BR}{SR}=\frac{BC}{ST}=\frac{RC}{RT} \). So \( \frac{x}{12}=\frac{10}{15} \). \( \frac{10}{15}=\frac{2}{3} \), so \( x = 12\times\frac{2}{3}=8 \). Wait, the handwritten 10.5 is wrong? Wait, maybe the height is \( SR = 12 \), base \( RT = 9 \), and the smaller triangle has base \( RC = 6 \), so the ratio of bases is \( \frac{6}{9}=\frac{2}{3} \), so the height of the smaller triangle (BR) should be \( 12\times\frac{2}{3}=8 \). But let's check with \( \frac{BR}{12}=\frac{6}{9} \), \( 9BR = 72 \), \( BR = 8 \). Alternatively, if we consider the other side: \( \frac{BR}{12}=\frac{10}{15} \), same result. So maybe the handwritten 10.5 is incorrect, and the correct answer is 8. Wait, maybe I misread the diagram. Let me re - examine: The vertical side is 12, horizontal from R to T is 9, R to C is 6, BC is 10, ST is 15. So \( \triangle BRC \) with base 6, hypotenuse 10, so by Pythagoras, \( BR=\sqrt{10^{2}-6^{2}}=\sqrt{100 - 36}=\sqrt{64}=8 \). Ah, right! So the length of \( BR \) is 8.

Answer:

8