We will have to use the distance formula in order to determine the lengths of each side of the triangle.
Distance formula: \(\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} }\)
Let's calculate AB first:
A (1, 7) and B (-2, 2)
A: x1 = 1 and y1 = 7
B: x2 = -2 and y2 = 2
so
\(\sqrt{(-2 - 1)^{2} + (2 - 7)^{2} }\)
\(\sqrt{(-3)^{2} + (-5)^{2} }\)
\(\sqrt{9 + 25 }\)
AB = \(\sqrt{34}\) or (rounded to the nearest tenth) ≈ 5.8
Now let's do BC:
B: x1 = -2 and y1 = 2
C: x2 = 4 and y2 = 2
So
\(\sqrt{(4 - -2)^{2} + (2 - 2)^{2} }\)
\(\sqrt{(6)^{2} + (0)^{2} }\)
BC = \(\sqrt{36 }\) or 6
Now let's do CA
C: x1 = 4 and y1 = 2
A: x2 = 1 and y2 = 7
So
\(\sqrt{(1 - 4)^{2} + (7 - 2)^{2} }\)
\(\sqrt{(-3)^{2} + (5)^{2} }\)
\(\sqrt{9 + 25}\)
CA = \(\sqrt{34}\) or (rounded to the nearest tenth) ≈ 5.8
So let's recap:
AB ≈ 5.8
BC = 6
CA ≈ 5.8
So AB and AC are the same length while BC is .2 units longer which means this is an isosceles triangle.
