Question

this figure shows $\triangle abc$. $\overline{bd}$ is the angle bisector of $\angle abc$. what is $ad$? enter your answer, as a fraction, in the box. \boxed{} units

Explanation:

Step1: Recall Angle Bisector Theorem

The Angle Bisector Theorem states that $\frac{AD}{DC} = \frac{AB}{BC}$

Step2: Define variables for segments

Let $AD = x$, so $DC = 6 - x$. Substitute known values: $AB=8$, $BC=10$.
$\frac{x}{6 - x} = \frac{8}{10}$

Step3: Simplify and solve for $x$

First simplify $\frac{8}{10} = \frac{4}{5}$, then cross-multiply:
$5x = 4(6 - x)$
$5x = 24 - 4x$
$5x + 4x = 24$
$9x = 24$
$x = \frac{24}{9} = \frac{8}{3}$

Answer:

$\frac{8}{3}$ units