Question

35
type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar.

in $\triangle abc$, $overline{ab} = overline{bc}$ and $mangle abc = 90^circ$. $d$ and $e$ are the midpoints of $overline{ab}$ and $overline{bc}$, respectively.

if the length of $overline{ca}$ is 9 units, the length of $overline{de}$ is \underline{\quad\quad} units and $mangle cab$ is \underline{\quad\quad} $^circ$.

Explanation:

Step1: Identify triangle type

$\triangle ABC$ is isosceles right triangle, $\overline{AB}=\overline{BC}$, $\angle ABC=90^\circ$

Step2: Apply Midsegment Theorem

$DE$ is midsegment, so $DE=\frac{1}{2}CA$
$DE=\frac{1}{2} \times 9 = \frac{9}{2}$

Step3: Calculate $\angle CAB$

In isosceles right triangle, base angles are equal:
$\angle CAB = \frac{180^\circ - 90^\circ}{2} = 45^\circ$

Answer:

The length of $\overline{DE}$ is $\frac{9}{2}$ units and $m\angle CAB$ is $45$ °.