Question

question
solve for ( x ). figures are not necessarily drawn to scale.
(figure with triangle qrs, segment ru = 3, us = 6, angle at q and angle at t are 54°, segment ts = 7, segment qt = x)

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle QRS \) and \( \triangle TUS \) are similar because they have two equal angles (the \( 54^\circ \) angle and the right angle, by AA similarity criterion).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \( \frac{QT + TS}{TS} = \frac{RU + US}{US} \). Let \( QT = x \), \( TS = 7 \), \( RU = 3 \), \( US = 6 \). Then \( \frac{x + 7}{7} = \frac{3 + 6}{6} \).

Step3: Solve the Proportion

Simplify the right - hand side: \( \frac{9}{6}=\frac{3}{2} \). So the equation becomes \( \frac{x + 7}{7}=\frac{3}{2} \). Cross - multiply: \( 2(x + 7)=3\times7 \). Expand: \( 2x+14 = 21 \). Subtract 14 from both sides: \( 2x=21 - 14=7 \). Divide by 2: \( x=\frac{7}{2}=3.5 \)? Wait, no, wait. Wait, the ratio of corresponding sides: \( \frac{QR}{TU}=\frac{RS}{US}=\frac{QS}{TS} \). Wait, \( RS=3 + 6 = 9 \), \( US = 6 \), \( TS = 7 \), \( QS=x + 7 \). So \( \frac{QS}{TS}=\frac{RS}{US}\), so \( \frac{x + 7}{7}=\frac{9}{6}=\frac{3}{2} \). Then \( 2(x + 7)=21 \), \( 2x+14 = 21 \), \( 2x=7 \), \( x = 3.5 \)? Wait, no, maybe I mixed up the sides. Wait, the two triangles: \( \triangle QRU \) and \( \triangle TUS \)? No, the vertical sides: \( RU = 3 \), \( US = 6 \), so the ratio of \( RU:US=3:6 = 1:2 \). The angle at Q is common (54 degrees) and the right angles, so the triangles are similar with a ratio of 1:2? Wait, no, \( RU = 3 \), \( US = 6 \), so the base of the big triangle is \( 3 + 6=9 \), the base of the small triangle is 6. The side \( TS = 7 \), and \( QS=x + 7 \). So the ratio of similarity is \( \frac{9}{6}=\frac{3}{2} \), so \( \frac{QS}{TS}=\frac{3}{2} \), so \( QS=\frac{3}{2}\times TS=\frac{3}{2}\times7 = 10.5 \), then \( x=QS - TS=10.5 - 7 = 3.5 \)? Wait, no, that can't be. Wait, maybe the ratio is \( \frac{RU}{US}=\frac{QT}{TS} \). \( RU = 3 \), \( US = 6 \), so \( \frac{3}{6}=\frac{x}{7} \), so \( \frac{1}{2}=\frac{x}{7} \), then \( x = 3.5 \). Yes, that's correct. Because the triangles are similar, the ratio of the segments on the base (RU:US) is equal to the ratio of the segments on the hypotenuse (QT:TS). So \( \frac{RU}{US}=\frac{QT}{TS} \), \( \frac{3}{6}=\frac{x}{7} \), simplify \( \frac{1}{2}=\frac{x}{7} \), cross - multiply \( x=\frac{7}{2}=3.5 \).

Answer:

\( x = 3.5 \) (or \( \frac{7}{2} \))