Question

Complete.

If$\overline{\mathit{R}\mathit{K}}$is both an altitude and a median of $\mathbf{\Delta }\mathit{R}\mathit{S}\mathit{T}$, then:

1. $\mathbf{\Delta }\mathit{R}\mathit{S}\mathit{K}\mathbf{\cong }\mathbf{\Delta }\mathit{R}\mathit{T}\mathit{K}$by $\underset{¯}{?}$
2. $\mathbf{\Delta }\mathit{R}\mathit{S}\mathit{T}$is $\mathit{a}\mathbf{\left(}\mathbf{n}\mathbf{\right)}\underset{\mathbf{¯}}{\mathbf{?}}$triangle
   
   

Short answer

a.$\mathrm{\Delta }RSK\cong \mathrm{\Delta }RTK$ by SAS Postulate.

b.$\mathrm{\Delta }RST$ is an isosceles triangle

Step-by-step solution

Part a. Step 1. Definition of median and altitude.

The altitude is perpendicular line segment from vertex to opposite side of thetriangle.

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

Part a. Step 2. Show that ΔRSK≅ΔRTK

Statement	Reason
1.$\overline{RK}\cong \overline{RK}$	Common line segment
2.$\angle RKS\cong \angle RKT$	Altitude divides adjacent angles into congruent angles
3.$\overline{SK}\cong \overline{TK}$	$K$is midpoint of $\overline{ST}$, so divide it into two congruent sides
4.$\mathrm{\Delta }RSK\cong \mathrm{\Delta }RTK$	SAS (Side-Angle-Side)Postulate

Part a. Step 3. Complete the given statement.

$\mathrm{\Delta }RSK\cong \mathrm{\Delta }RTK$by SAS Postulate.

Part b. Step 1. Definition of Isosceles triangle.

A triangle is isosceles when two of its sides are congruent

Part b. Step 2. Show that SR¯≅TR¯

Statement	Reason
1.$\overline{RK}\cong \overline{RK}$	Common line segment
2.$\angle RKS\cong \angle RKT$	Altitude divides adjacent angles into congruent angles
3.$\overline{SK}\cong \overline{TK}$	$K$is midpoint of $\overline{ST}$, so divide it into two congruent sides
4.$\mathrm{\Delta }RSK\cong \mathrm{\Delta }RTK$	SAS (Side-Angle-Side)Postulate
5.$\overline{SR}\cong \overline{TR}$	Corresponding parts of congruent triangle

Part b. Step 3. Complete the given statement.

$\mathrm{\Delta }RST$is an isosceles triangle