Answer
scalene
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 $C(-2, 0)$.
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:
$AB = \sqrt {(2)^2 + (-4)^2}$
Evaluate the exponents:
$AB = \sqrt {4 + 16}$
Add what is underneath the radical:
$AB = \sqrt {20}$
Rewrite $20$ as the product of a perfect square and another number:
$AB = \sqrt {4 • 5}$
Take the square root of $4$ to simplify the radical:
$AB = 2 \sqrt {5}$
Let's look at the next side, $BC$:
$BC = \sqrt {(-2 - 3)^2 + (0 - (-1))^2}$
Simplify within the parentheses:
$BC = \sqrt {(-5)^2 + (1)^2}$
Evaluate the exponents:
$BC = \sqrt {25 + 1}$
Add what is underneath the radical:
$BC = \sqrt {26}$
Let's look at $CA$:
$CA = \sqrt {(-2 - 1)^2 + (0 - 3)^2}$
Simplify within the parentheses:
$CA = \sqrt {(-3)^2 + (-3)^2}$
Evaluate the exponents:
$CA = \sqrt {9 + 9}$
Add what is underneath the radical:
$CA = \sqrt {18}$
Rewrite $18$ as the product of a perfect square and another number:
$CA = \sqrt {9 • 2}$
Take the square root of $9$ to simplify the radical:
$CA = 3 \sqrt {2}$
All three sides have different lengths; therefore, this triangle is scalene.
