Question

in the diagram below, \\(\overline{ts}\\) is parallel to \\(\overline{al}\\).
find the length of \\(\overline{bl}\\).
if entering your answer as a decimal, round your final answer to the nearest hundredth.
\\(bl = \square\\) units

Explanation:

Step1: Identify Similar Triangles

Since \(\overline{TS} \parallel \overline{AL}\), by the Basic Proportionality Theorem (Thales' theorem), \(\triangle BTS \sim \triangle BAL\). So, the ratios of corresponding sides are equal: \(\frac{TS}{AL}=\frac{BS}{BL}\). Let \(BL = x\), then \(BS=x - 5.1\).

Step2: Set Up Proportion

Substitute the known values: \(\frac{7.75}{12}=\frac{x - 5.1}{x}\).

Step3: Cross - Multiply

Cross - multiplying gives \(7.75x=12(x - 5.1)\).

Step4: Expand and Solve

Expand the right - hand side: \(7.75x = 12x-61.2\).
Subtract \(7.75x\) from both sides: \(0 = 12x-7.75x - 61.2\).
Simplify: \(0 = 4.25x-61.2\).
Add \(61.2\) to both sides: \(4.25x=61.2\).
Divide both sides by \(4.25\): \(x=\frac{61.2}{4.25}=14.4\).

Answer:

\(14.4\)