Question

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

diagram of triangle with points a, b, c, d; ac = 5.5, cd = 4.3, db = 4.1, angles at a are equal

Explanation:

Step1: Identify the Theorem

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

Step2: Substitute the Known Values

We know \( AC = 5.5 \), \( CD = 4.3 \), and \( DB = 4.1 \). Let \( AB = x \). Substituting into the theorem: \( \frac{5.5}{x}=\frac{4.3}{4.1} \).

Step3: Solve for \( x \)

Cross - multiply to get \( 4.3x=5.5\times4.1 \). First, calculate \( 5.5\times4.1 = 22.55 \). Then, \( x=\frac{22.55}{4.3}\approx5.2 \).

Answer:

\( 5.2 \)