Question

$\angle dac = \angle bad$. what is the length of $\overline{ac}$? round to one decimal place. (triangle with points a, b, c, d. ac is labeled?, ad divides angle bac into two equal angles (θ). cd = 2.5, db = 3.8, ab = 6.8)

Explanation:

Step1: Identify the Angle Bisector Theorem

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

Step2: Determine the lengths of \( CD \), \( DB \), and \( AB \)

From the diagram, \( CD = 2.5 \), \( DB=3.8 \), and \( AB = 6.8 \). Let \( AC=x \).

Step3: Apply the Angle Bisector Theorem formula

Substitute the known values into the formula \( \frac{AC}{AB}=\frac{CD}{DB} \), we get \( \frac{x}{6.8}=\frac{2.5}{3.8} \).

Step4: Solve for \( x \)

Cross - multiply to solve for \( x \): \( x=\frac{2.5\times6.8}{3.8} \).
First, calculate the numerator: \( 2.5\times6.8 = 17 \).
Then, divide by the denominator: \( x=\frac{17}{3.8}\approx4.5 \).

Answer:

\( 4.5 \)