Question

Identify the triangle PQR as isosceles​ (two sides of equal​ length), equilateral​ (three sides of equal​ length), or a scalene triangle​ (three sides of different​ lengths). ​P(negative 4​,9​), ​Q(0​,8​), and ​R(negative 3​,5​)

Answer 1

Isosceles

Explanation:

In order to figure what type of triangle this is out, we need to start by plotting the given points. That will help us visualize the triangle better and see if our conclusions make sense. (See attached picture).

Once we got the triangle, the strategy to follow is to use the distance between two points formula to see what the measurement of each side of the triangle is. This will help us determine if 2, 3 or none of the sides of the tirangle are the same.

The distance formula is the following:

`distance=\sqrt{(x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2)}`

so now we can find the desired distances, let's start with the distance between P and Q:

`|PQ|=\sqrt{(-4-0)^(2)+(9-8)^(2)}`

which yields:

`|PQ|=√(17)`

next, let's find the distance between P and R:

`|PR|=\sqrt{(-4-(-3))^(2)+(9-5)^(2)}`

which yields:

`|PR|=√(17)`

and finally the distance between Q and R:

`|QR|=\sqrt{(0-(-3))^(2)+(8-5)^(2)}`

which yields:

`|QR|=3√(2)`

As you may see from the result, only two of the three sides are the same, |PQ| and |PR|, so this will be an Isosceles triangle.