Question

find y.
u to v is a horizontal line with an arrow. u to s has length 18 from u to t and 4y+16 from t to s. v to s has length 20 from v to r and 40 from r to s. t to r is a horizontal line with an arrow. find y. y = submit

Explanation:

Step1: Identify Similar Triangles

Since \( UV \parallel TR \), triangles \( \triangle USV \) and \( \triangle TSR \) are similar by the Basic Proportionality Theorem (Thales' theorem). In similar triangles, the ratios of corresponding sides are equal. So, \(\frac{UT}{TS}=\frac{VR}{RS}\)? Wait, no, actually, the ratio of \( UT \) to \( US \) should equal the ratio of \( VR \) to \( VS \). Wait, \( US = UT + TS = 18+(4y + 16)=4y + 34 \), and \( VS = VR + RS = 20 + 40 = 60 \). Wait, no, maybe better: \( \frac{UT}{US}=\frac{VR}{VS} \)? Wait, no, \( UT = 18 \), \( TS = 4y + 16 \), \( VR = 20 \), \( RS = 40 \). Since \( UV \parallel TR \), the triangles \( \triangle UVS \) and \( \triangle TRS \) are similar, so the ratio of \( UT \) to \( TS \) is equal to the ratio of \( VR \) to \( RS \)? Wait, no, actually, the segments on \( US \) and \( VS \): \( \frac{UT}{TS}=\frac{VR}{RS} \)? Wait, \( UT = 18 \), \( TS = 4y + 16 \), \( VR = 20 \), \( RS = 40 \). So \( \frac{18}{4y + 16}=\frac{20}{40} \). Simplify \( \frac{20}{40}=\frac{1}{2} \). So \( \frac{18}{4y + 16}=\frac{1}{2} \).

Step2: Solve the Proportion

Cross - multiply: \( 1\times(4y + 16)=18\times2 \)
\( 4y+16 = 36 \)
Subtract 16 from both sides: \( 4y=36 - 16 \)
\( 4y = 20 \)
Divide both sides by 4: \( y=\frac{20}{4}=5 \)

Answer:

\( y = 5 \)