Question

If A = (1, 4) B = (3, 1) and C = (6, 3), classify the the following triangle as either equilateral

Answer 1

To find the answer, we must first educate ourselves on the types of triangles.

Equilateral- All 3 sides of the triangle are the same length.
Isosceles- 2 out of the 3 sides are the same length.
Scalene- No sides of the triangle are the same length.

With this in mind, we should plot our given points on a graph, and connect the points to create our triangle.

Now, we will need to use the distance formula for each side to find the lengths. The distance formula is:

$\sqrt{(x_{2}- x_{1})^{2}+ (y_{2}- y_{1})^{2} }$   where the "x's and y's refer to the coordinates of the points.

Now that we are familiar with the distance formula, we can apply it by substituting in our values and solving.

Distance between (1, 4) and (3, 1)
$\sqrt{(x_{2}- x_{1})^{2}+ (y_{2}- y_{1})^{2} }$
$\sqrt{(3-1)^{2}+(1-4)^{2} }$
$\sqrt{(2)^{2}+(-3)^{2} }$
$\sqrt{13}$

Distance between (3, 1) and (6, 3)
$\sqrt{(x_{2}- x_{1})^{2}+ (y_{2}- y_{1})^{2} }$
$\sqrt{(6-3)^{2}+(3-1)^{2}}$
$\sqrt{(3)^{2}+(2)^{2}}$
$\sqrt{13}$

Distance between (6, 3) and (1, 4)
$\sqrt{(x_{2}- x_{1})^{2}+ (y_{2}- y_{1})^{2} }$
$\sqrt{(1-6)^{2}+(4-3)^{2}}$
$\sqrt{(-5)^{2}+(1)^{2}}$
$\sqrt{26}$

Using the distance formula, we can see that the 3 lengths are $\sqrt{13}$, $\sqrt{13}$, and $\sqrt{26}$. Now, let's go back to our different types of triangles and find out what kind this is.

Since 2 of our side lengths are the same, this is an isosceles triangle.

Before we can be sure about this answer, we need to check the graph we made earlier and simply look at it to approximate if our answer is correct. When we do this, we will see that our answer is indeed correct.

Below, I have attached a graph with our points plotted.