Question

What is the perimeter of a triangle ABC on a graph? A (6,1) B (2,7) C (-3,-5)

Answer 1

`13 + 5sqrt13`

Explanation:

Let's see what this triangle looks like.

I used desmos.com to make the graph; it's a great free online graphing calculator!

Anyway, let's use the Pythagorean theorem to find each of the sides. Let's start with the side connecting (-3, -5) and (2, 7). If you go "over" 5 along the x-axis, and "up" 12 along the y-axis, you get from (-3, -5) to (2, 7). So, this side can be thought of as the hypotenuse of a right triangle with legs of 5 and 12.

`5^2 + 12^2 = x^2`
`169 = x^2`
`13 = x`

So this side has length 13. Now let's find the length of the side connecting (2, 7) and (6, 1). To get from (2, 7) to (6, 1), you go "down" 6 and "over" 4. So, this side is the hypotenuse of a right triangle with sides of 6 and 4.

`6^2 + 4^2 = x^2`
`52 = x^2`
`2sqrt(13)=x`

So this side has length `2sqrt13`. One last side (the one from (-3, -5) to (6, 1)). To get from (-3, -5) to (6, 1) you go "over" 9 and "up" 6. So, this side is the hypotenuse of a right triangle with sides of 9 and 6.

`9^2+6^2=x^2`
`117=x^2`
`3sqrt13=x`

So this side has length `3sqrt13`.

This means the total perimeter is 13 + `2sqrt13` + `3sqrt13` or `13 + 5sqrt13`.