Chapter 4.7, Problem 4CE

a.

To determine

To complete: the blank in the given statement.

Answer to Problem 4CE

Side-Angle-Side postulate

Explanation of Solution

Given information:

The given statement is “If $\overline{RK}$ is the both altitude and median of $\Delta RST.$ , then

  $\Delta RSK\cong \Delta RTK$ by $\underset{\_}{}$ ”.

The given figure is as follows:

  

Concept used:

• The altitude is perpendicular line segment from vertex to opposite side of the

triangle.

• The median is a line segment from vertex to midpoint of the opposite side of the

triangle.

• Side-Angle-Sidepostulate: fortwo triangles, the two corresponding sides and

angle between them are congruent. Then the two triangles are congruent.

Consider $\overline{RK}$ is the both altitude and median of the triangle $\Delta RST.$

  $\begin{array}{l}\overline{RK}\perp \overline{ST}\\ \overline{SK}\cong \overline{KT}\end{array}$

By Side angle side postulate,

The two triangles, $\Delta RSK,\Delta RTK$ are congruent. Since

The corresponding sides $\overline{SK}\cong \overline{KT}$ , common side $\overline{RK}$ and angles between them

  $\angle SKR\cong \angle RKT$ are congruent.

The blank is filled with term Side angle Side.

b.

To determine

To complete: the blank in the given statement.

Answer to Problem 4CE

Isosceles

Explanation of Solution

Given information:

The given statement is “If $\overline{RK}$ is the both altitude and median of $\Delta RST.$ , then

  $\Delta RST$ is $\underset{\_}{}$ triangle.”

The given figure is as follows:

  

Consider the given triangle $\Delta RST.$

From part a,

The two triangles $\Delta RSK,\Delta RTK$ are congruent by Side angle Side congruence criteria.

So, the corresponding angles are congruent.

  $\overline{RS}=\overline{RT}$ .

Since $\Delta RST$ has two equal sides, it is an isosceles triangle.

The blank is filled with term isosceles.