Chapter 4.5, Problem 19WE

To determine

To prove: The segments are drawn from the midpoints of the legs of an isosceles triangle $△ABC$ perpendicular to the base and the segments are congruent.

Explanation of Solution

Given information:

System information:isosceles triangle $△ABC$ with equal side $AB$ and $AC$

Calculation:

Consider an isosceles triangle $△ABC$

  

Now,

Draw segment $MP\perp BC$ where $\text{M}$ is the midpoint of $AB$ and draw a segment $\overrightarrow{NQ}\perp \overrightarrow{BC}$ where $N$ is the midpoint of $AC$ .

  

Now,

To proof that $\overline{MP}\cong \overline{NQ}$

So,

The M midpoint of $AB$ and $MP\perp BC$

The N midpoint of $AB$ and $\overrightarrow{NQ}\perp \overrightarrow{BC}$

The two columns proof is stated below:

Statements Reason

1. $AB=AC$ Given

2. $AM=MB$ $M$ is the midpoint of $AB$

3. $AN=NC$ $N$ is the midpoint of $AC$

4. $\begin{array}{l}△MBP\text{ and }\angle NCQ\\ \angle MPB\cong \angle NQC\end{array}$ By $MP\perp BC$ and $\overrightarrow{NQ}\perp \overrightarrow{BC}$

5. $\overline{MB}\cong \overline{NC}$ by Using 1, 2, 3

6. $\angle B=\angle C$ Base angle of an isosceles triangle

7. $△MBP\cong \angle NCQ$ by AAS

8. $\overline{MP}\cong \overline{NQ}$ Corresponding sides of congruent triangles