Question

triangle ABC has vertices A (4,12) B(6,6) and C (18,10). Classify triangle ABC by side length. is it a right triangle?

Answer 1

So,

To classify this triangle, we're going to find the distance between each vertex. Then, we're going to compare all these measures and analyze.

First, let's begin with the side AB.

Let me upload an image with the points given:

What we could do to find the length of AB, is to count the horizontal and vertical units between both points. Then, as you can see, there's a triangle formed and the side AB is the hypotenuse, so we could apply the pithagorean theorem as follows:

`\begin{gathered} AB=\sqrt[]{6^2+2^2} \\ AB=\sqrt[]{40} \end{gathered}`

We are going to do the same process with the other points:

Let's go with AC:

Applying the pythagorean theorem, we got that:

`\begin{gathered} AC=\sqrt[]{14^2+2^2} \\ AC=\sqrt[]{200} \end{gathered}`

And, finally, we're going to find the lenght of BC:

That is:

`\begin{gathered} BC=\sqrt[]{(12)^2+4^2} \\ BC=\sqrt[]{160} \end{gathered}`

Then, we got that the lengths of each side of the triangle formed by three points are:

`\begin{gathered} AB=\sqrt[]{40} \\ AC=\sqrt[]{200} \\ BC=\sqrt[]{160} \end{gathered}`

As you can see, the three lengths are different. So, the following triangle:

As all sides are different, this is a scalene triangle.

To check that it is a right triangle, notice that if we make:

`\begin{gathered} (\sqrt[]{40})^2+(\sqrt[]{160})^2=(\sqrt[]{200})^2 \\ 200=200 \end{gathered}`

So as the sides satisfy the pythagorean theorem, the triangle is actually a right triangle. You could also notice that the angle between AB and BC is 90°, so that's other reason for the triangle to be a right triangle.

It is a right scalene triangle.