Question

solve for x. round your answer to the nearest tenth if necessary.

Explanation:

Step1: Identify Similar Triangles

Since \( TU \parallel QR \), triangles \( STU \) and \( SQR \) are similar by the Basic Proportionality Theorem (Thales' theorem).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{ST}{SQ}=\frac{SU}{SR}\). Let \( ST = x \), then \( SQ=x + 14 \), \( SU = 23.3 \), and \( SR=23.3 + 11.7=35 \). The proportion becomes \(\frac{x}{x + 14}=\frac{23.3}{35}\).

Step3: Cross - Multiply and Solve

Cross - multiply: \( 35x=23.3(x + 14) \).
Expand the right - hand side: \( 35x=23.3x+23.3\times14 \).
Calculate \( 23.3\times14 = 326.2 \), so \( 35x=23.3x + 326.2 \).
Subtract \( 23.3x \) from both sides: \( 35x-23.3x=326.2 \).
Simplify the left - hand side: \( 11.7x=326.2 \).
Solve for \( x \): \( x=\frac{326.2}{11.7}\approx27.9 \).

Answer:

\( x\approx27.9 \)