Answer
isosceles
Work Step by Step
We use the distance formula to determine what type of triangle is pictured.
The vertices of the triangle are $A(1, 3)$, $B(3, 1)$, and $C(-2, -2)$.
The distance formula is given by the following formula:
$d = \sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Let's determine the lengths of the different sides of the triangle. We'll look at $AB$ first:
$AB = \sqrt {(3 - 1)^2 + (1 - 3)^2}$
Simplify within the parentheses:
$AB = \sqrt {(2)^2 + (-2)^2}$
Evaluate the exponents:
$AB = \sqrt {4 + 4}$
Add what is underneath the radical:
$AB = \sqrt {8}$
Rewrite $8$ as the product of a perfect square and another number:
$AB = \sqrt {4 • 2}$
Take the square root of $4$ to simplify the radical:
$AB = 2 \sqrt {2}$
Let's look at the next side, $BC$:
$BC = \sqrt {(-2 - 3)^2 + (-2 - 1)^2}$
Simplify within the parentheses:
$BC = \sqrt {(-5)^2 + (-3)^2}$
Evaluate the exponents:
$BC = \sqrt {25 + 9}$
Add what is underneath the radical:
$BC = \sqrt {34}$
Let's look at $CA$:
$CA = \sqrt {(-2 - 1)^2 + (-2 - 3)^2}$
Simplify within the parentheses:
$CA = \sqrt {(-3)^2 + (-5)^2}$
Evaluate the exponents:
$CA = \sqrt {9 + 25}$
Add what is underneath the radical:
$CA = \sqrt {34}$
$BC = CA$; therefore, this triangle is isosceles.
