Question

∠dac = ∠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 \(\angle BAC\) (where \(\angle DAC=\angle BAD\)), 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: \(5.1x=5.7\times3.2\).
First, calculate \(5.7\times3.2 = 18.24\).
Then, \(x=\frac{18.24}{5.1}\).
\(x=\frac{1824}{510}=\frac{304}{85}\approx3.6\).

Answer:

\(3.6\)