Question

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

image of a triangle with vertices a, b, c; ac = 5.1, cd = 3.2, ab = 5.7, angles at a (∠dac and ∠bad) are equal (marked as θ)

Explanation:

Step1: Identify the Theorem

Since \( \angle DAC = \angle BAD \), \( AD \) is the angle bisector of \( \angle CAB \). By the Angle Bisector Theorem, \( \frac{BD}{DC}=\frac{AB}{AC} \).

Step2: Define Variables

Let \( BD = x \), then \( DC = 3.2 \), \( AB = 5.7 \), \( AC = 5.1 \).

Step3: Apply the Theorem

Substitute into the ratio: \( \frac{x}{3.2}=\frac{5.7}{5.1} \).

Step4: Solve for \( x \)

Multiply both sides by \( 3.2 \): \( x=\frac{5.7\times3.2}{5.1} \).
Calculate numerator: \( 5.7\times3.2 = 18.24 \).
Divide by \( 5.1 \): \( x=\frac{18.24}{5.1}\approx3.6 \).

Answer:

\( 3.6 \)