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 = \boxed{\quad}\\) units

Explanation:

Step1: Set up proportion via similarity

Since $\overline{TS} \parallel \overline{AL}$, $\triangle BTS \sim \triangle BAL$. Let $BL = x$, so $BS = x - 5.1$. The proportion of corresponding sides is:
$$\frac{TS}{AL} = \frac{BS}{BL}$$
Substitute values: $\frac{7.75}{12} = \frac{x - 5.1}{x}$

Step2: Cross-multiply to solve for x

Cross-multiply to eliminate fractions:
$$7.75x = 12(x - 5.1)$$
Expand right side: $7.75x = 12x - 61.2$

Step3: Isolate x term

Rearrange to collect x terms on one side:
$$12x - 7.75x = 61.2$$
Calculate left side: $4.25x = 61.2$

Step4: Solve for x

Divide both sides by 4.25:
$$x = \frac{61.2}{4.25}$$

Answer:

14.40 units