Question

type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar. in △abc, $overline{ab}=overline{bc}$ and m∠abc = 90°. d and e are the mid - points of $overline{ab}$ and $overline{bc}$, respectively. if the length of $overline{ca}$ is 9 units, the length of $overline{de}$ is units and m∠cab is °.

Explanation:

Step1: Recall the mid - point theorem for right - angled triangle

In a right - angled triangle, the line segment joining the mid - points of two sides is half the length of the hypotenuse. Here, in right - angled triangle $\triangle ABC$ with $\angle ABC = 90^{\circ}$, and $D$ and $E$ are mid - points of $AB$ and $BC$ respectively, and $CA$ is the hypotenuse.

Step2: Calculate the length of $DE$

Since $DE=\frac{1}{2}CA$, and $CA = 9$ units, then $DE=\frac{9}{2}=4.5$ units.

Step3: Find the measure of $\angle CAB$

In $\triangle ABC$, $AB = BC$ and $\angle ABC=90^{\circ}$. So, $\triangle ABC$ is an isosceles right - angled triangle. Using the angle - sum property of a triangle ($\angle ABC+\angle BAC+\angle BCA = 180^{\circ}$) and since $\angle BCA=\angle BAC$ (because $AB = BC$), we have $90^{\circ}+2\angle BAC=180^{\circ}$, then $2\angle BAC = 90^{\circ}$, and $\angle BAC = 45^{\circ}$.

Answer:

$4.5$; $45$