Question

if rm = 12, lr = 9, and mq = 16, find tq. image not set to scale. none of these answers are correct. 12.48 3.72 8.63 6.86

Explanation:

Step1: Find ML length

First, we find the length of \( ML \). Since \( LR = 9 \) and \( RM = 12 \), we use the segment addition postulate: \( ML=LR + RM \). So \( ML = 9 + 12=21 \).

Step2: Set up proportion (similar triangles)

Triangles \( MRT \) and \( MLQ \) are similar (by AA similarity, as \( \angle MTR=\angle MQL = 90^\circ \) and they share \( \angle TMR=\angle QML \)). So the ratios of corresponding sides are equal: \( \frac{RM}{ML}=\frac{MT}{MQ} \). Let \( TQ = x \), then \( MT = MQ - TQ=16 - x \). Substituting the known values: \( \frac{12}{21}=\frac{16 - x}{16} \).

Step3: Solve the proportion

Cross - multiply: \( 12\times16=21\times(16 - x) \)
\( 192 = 336-21x \)
Add \( 21x \) to both sides: \( 21x + 192=336 \)
Subtract 192 from both sides: \( 21x=336 - 192=144 \)
Then \( x=\frac{144}{21}\approx6.86 \)

Answer:

6.86