Question

in the diagram below, \\(\overline{uv}\\) is parallel to \\(\overline{rs}\\). if \\(uv\\) is 5 less than \\(tv\\), \\(ts = 45\\), and \\(rs = 36\\), find the length of \\(\overline{tv}\\). figures are not necessarily drawn to scale. state your answer in simplest radical form, if necessary.

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{UV} \parallel \overline{RS} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle TUV \sim \triangle TRS \) (AA similarity, as corresponding angles are equal).

Step2: Set Variables for Lengths

Let \( TV = x \). Then \( UV = x - 5 \) (given \( UV \) is 5 less than \( TV \)). Also, \( TS = 45 \), so \( VS = TS - TV = 45 - x \).

Step3: Apply Similar Triangles Ratio

For similar triangles \( \triangle TUV \) and \( \triangle TRS \), the ratio of corresponding sides is equal. So \( \frac{UV}{RS} = \frac{TV}{TS} \).
Substitute the known values: \( \frac{x - 5}{36} = \frac{x}{45} \).

Step4: Solve the Proportion

Cross - multiply: \( 45(x - 5)=36x \).
Expand the left - hand side: \( 45x-225 = 36x \).
Subtract \( 36x \) from both sides: \( 45x-36x-225=36x - 36x \), which gives \( 9x-225 = 0 \).
Add 225 to both sides: \( 9x=225 \).
Divide both sides by 9: \( x = 25 \).

Answer:

The length of \( \overline{TV} \) is 25.