Question

Show that the triangle with vertices$A\left(-3,4\right),M\left(3,1\right),\text{and}Y\left(0,2\right)$is an isosceles triangle.

Short answer

Triangle AMY is an isosceles triangle because $AM=AY$.

Step-by-step solution

Step-1 – Given

The given vertices are $A\left(-3,4\right),M\left(3,1\right),\text{and}Y\left(0,2\right)$

Step-2 – To determine

We have to show that the triangle is an isosceles triangle.

Step-3 – Proof

An isosceles triangle is a triangle with two sides of equal length.

We will use the distance formula to show that$AM=AY$$AM=AY$

Using the distance formula, the length AM is:

$\begin{array}{l}AM=\sqrt{{\left(3-\left(-3\right)\right)}^2+{\left(1-4\right)}^2}\\ AM=\sqrt{{\left(3+3\right)}^2+{\left(-3\right)}^2}\\ AM=\sqrt{{\left(6\right)}^2+{\left(-3\right)}^2}\\ AM=\sqrt{36+9}\\ AM=\sqrt{45}\end{array}$

And, the length AY is:

$\begin{array}{l}AY=\sqrt{{\left(0-\left(-3\right)\right)}^2+{\left(-2-4\right)}^2}\\ AY=\sqrt{{\left(0+3\right)}^2+{\left(-2-4\right)}^2}\\ AY=\sqrt{{\left(3\right)}^2+{\left(-6\right)}^2}\\ AY=\sqrt{9+36}\\ AY=\sqrt{45}\end{array}$

Hence, we have shown that$AM=AY$ and that means the triangle AMY is an isosceles triangle.