Question

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

Explanation:

Step1: Identify the Theorem

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

Step2: Determine Segment Lengths

From the diagram, \( CD = 4.3 \), \( DB = 4 \), \( AB = 5.1 \), and let \( AC = x \).

Step3: Apply the Angle Bisector Theorem

Substitute into the theorem: \( \frac{x}{5.1}=\frac{4.3}{4} \).

Step4: Solve for \( x \)

Multiply both sides by \( 5.1 \): \( x=\frac{4.3\times5.1}{4} \). Calculate \( 4.3\times5.1 = 21.93 \), then \( x=\frac{21.93}{4}=5.4825 \). Round to one decimal place: \( x\approx5.5 \).

Answer:

\( 5.5 \)