Question

you might need: calculator
∠dac = ∠bad.
what is the length of \\(\overline{ac}\\)?
round to one decimal place.

(there is a triangle diagram with points a, b, c, d. ad is an angle bisector, with ∠dac = ∠bad = θ. lengths: ab = 6.2, bd = 3.7, cd = 2.5, and ac is unknown (marked as ?).)

Explanation:

Step1: Apply Angle Bisector Theorem

The Angle Bisector Theorem states that if a bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. So, $\frac{AC}{AB}=\frac{CD}{DB}$. Let $AC = x$, $AB = 6.2$, $CD = 2.5$, $DB = 2.7$. Then $\frac{x}{6.2}=\frac{2.5}{2.7}$.

Step2: Solve for \( x \)

Cross - multiply: $2.7x=6.2\times2.5$. Calculate $6.2\times2.5 = 15.5$. Then $x=\frac{15.5}{2.7}\approx5.7$ (rounded to one decimal place).

Answer:

\( 5.7 \)