Question

$\angle dac = \angle bad$.
what is the length of $\overline{bd}$?
round to one decimal place.

Explanation:

Step1: Apply the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if \(AD\) is the angle bisector of \(\angle BAC\) (\(\angle DAC=\angle BAD\)), then \(\frac{BD}{DC}=\frac{AB}{AC}\).
Let \(BD = x\), then \(DC = 3.2\), \(AB = 5.7\), and \(AC = 5.1\).
So, \(\frac{x}{3.2}=\frac{5.7}{5.1}\).

Step2: Solve for \(x\)

Cross - multiply the equation \(\frac{x}{3.2}=\frac{5.7}{5.1}\) to get \(5.1x=3.2\times5.7\).
First, calculate \(3.2\times5.7 = 18.24\).
Then, \(x=\frac{18.24}{5.1}\).
\(x = 3.6\).

Answer:

\(3.6\)