Chapter 4.8, Problem 20PPS

To determine

To prove:

The coordinate proof for the following statement:

‘The three segments joining the midpoints of the sides of an isosceles triangle form another isosceles triangle.

Explanation of Solution

Given information:

Type of triangle: Isosceles triangle Proof:

Here, first draw triangle $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}$ on a coordinate plane. Then need to estimate the midpoints $\mathrm{R},\mathrm{S},\mathrm{T}$ of the three sides and show that two of the three segments connecting the midpoints have the same length.

  

Start by placing a vertex at the origin and label it $\mathrm{A}$ . Then place point $\mathrm{B}$ on the $\mathrm{x}$ axis at point $\left(2\mathrm{a},0\right)$ .

Here, the type of triangle is given as isosceles so, the $\mathrm{x}$ -coordinate of point $\mathrm{C}$ is halfway between $0$ and $2\mathrm{a}$ or $\mathrm{a}$ .

Now, find the midpoints as:

  $\begin{array}{l}\mathrm{R}=\left(\frac{\mathrm{a}+0}{2},\frac{\mathrm{b}+0}{2}\right)=\left(\frac{\mathrm{a}}{2},\frac{\mathrm{b}}{2}\right)\\ \mathrm{S}=\left(\frac{\mathrm{a}+2\mathrm{a}}{2},\frac{\mathrm{b}+0}{2}\right)=\left(\frac{3\mathrm{a}}{2},\frac{\mathrm{b}}{2}\right)\\ \mathrm{R}=\left(\frac{2\mathrm{a}+0}{2},\frac{0+0}{2}\right)=\left(\mathrm{a},0\right)\end{array}$

Use the distance formula to find the side lengths of $\mathrm{\Delta }\mathrm{R}\mathrm{S}\mathrm{T}$ :

  $\begin{array}{l}\mathrm{R}\mathrm{T}=\sqrt{{\left(\frac{\mathrm{a}}{2}-\mathrm{a}\right)}^2+{\left(\frac{\mathrm{b}}{2}-\mathrm{a}\right)}^2}=\sqrt{{\left(\frac{\mathrm{a}}{2}\right)}^2+{\left(\frac{\mathrm{b}}{2}\right)}^2}\\ \mathrm{S}\mathrm{T}=\sqrt{{\left(\frac{3\mathrm{a}}{2}-\mathrm{a}\right)}^2+{\left(\frac{\mathrm{b}}{2}-\mathrm{a}\right)}^2}=\sqrt{{\left(\frac{\mathrm{a}}{2}\right)}^2+{\left(\frac{\mathrm{b}}{2}\right)}^2}\end{array}$

Therefore,

  $\mathrm{R}\mathrm{T}\cong \mathrm{S}\mathrm{T}$

There is no need to estimate the side length of $\mathrm{R}\mathrm{S}$ since these two sides already have equal measures. Therefore, by definition two sides same of the triangle, $\mathrm{\Delta }\mathrm{R}\mathrm{S}\mathrm{T}$ is isosceles triangle.

Hence proved.