Question

∠dac = ∠bad.
what is the length of \overline{ab}?
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 \(\angle DAC=\angle BAD = \theta\)), then \(\frac{AB}{AC}=\frac{BD}{DC}\).
Given \(AC = 5.6\), \(DC=4.3\), and \(BD = 4\).
Let \(AB=x\). Substitute the values into the formula: \(\frac{x}{5.6}=\frac{4}{4.3}\).

Step2: Solve for \(x\)

Cross - multiply the equation \(\frac{x}{5.6}=\frac{4}{4.3}\) to get \(4.3x=5.6\times4\).
First, calculate \(5.6\times4 = 22.4\). Then the equation becomes \(4.3x = 22.4\).
Solve for \(x\) by \(x=\frac{22.4}{4.3}\).
Using a calculator, \(x=\frac{22.4}{4.3}\approx5.2\).

Answer:

\(5.2\)