Question

1. a) on a grid, draw the triangle with each set of vertices. classify each triangle as scalene, isosceles, or equilateral.

i) a(-5, 5), b(2, 1), c(-6, 0)
ii) g(-2, -3), h(9, 1), j(7, 7)

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate side - lengths for first set of vertices A(-5,5), B(2,1), C(-6,0)

For side AB:
$d_{AB}=\sqrt{(2 + 5)^2+(1 - 5)^2}=\sqrt{49 + 16}=\sqrt{65}$
For side BC:
$d_{BC}=\sqrt{(-6 - 2)^2+(0 - 1)^2}=\sqrt{64 + 1}=\sqrt{65}$
For side AC:
$d_{AC}=\sqrt{(-6 + 5)^2+(0 - 5)^2}=\sqrt{1+25}=\sqrt{26}$
Since two sides ($AB$ and $BC$) are equal, the triangle with vertices A(-5,5), B(2,1), C(-6,0) is isosceles.

Step3: Calculate side - lengths for second set of vertices G(-2,-3), H(9,1), J(7,7)

For side GH:
$d_{GH}=\sqrt{(9 + 2)^2+(1 + 3)^2}=\sqrt{121+16}=\sqrt{137}$
For side HJ:
$d_{HJ}=\sqrt{(7 - 9)^2+(7 - 1)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10}$
For side GJ:
$d_{GJ}=\sqrt{(7 + 2)^2+(7 + 3)^2}=\sqrt{81 + 100}=\sqrt{181}$
Since all three sides have different lengths, the triangle with vertices G(-2,-3), H(9,1), J(7,7) is scalene.

Answer:

The triangle with vertices A(-5,5), B(2,1), C(-6,0) is isosceles. The triangle with vertices G(-2,-3), H(9,1), J(7,7) is scalene.