Question

6 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 part (a)

Given $A(-5,5)$, $B(2,1)$, $C(-6,6)$.
For side $AB$:
$x_1=-5,y_1 = 5,x_2=2,y_2 = 1$
$AB=\sqrt{(2+5)^2+(1 - 5)^2}=\sqrt{49 + 16}=\sqrt{65}$
For side $BC$:
$x_1=2,y_1 = 1,x_2=-6,y_2 = 6$
$BC=\sqrt{(-6 - 2)^2+(6 - 1)^2}=\sqrt{64+25}=\sqrt{89}$
For side $AC$:
$x_1=-5,y_1 = 5,x_2=-6,y_2 = 6$
$AC=\sqrt{(-6 + 5)^2+(6 - 5)^2}=\sqrt{1+1}=\sqrt{2}$
Since all side - lengths are different, the triangle is scalene.

Step3: Calculate side - lengths for part (b)

Given $G(-2,-3)$, $H(9,1)$, $J(7,7)$
For side $GH$:
$x_1=-2,y_1=-3,x_2=9,y_2 = 1$
$GH=\sqrt{(9 + 2)^2+(1+3)^2}=\sqrt{121 + 16}=\sqrt{137}$
For side $HJ$:
$x_1=9,y_1 = 1,x_2=7,y_2 = 7$
$HJ=\sqrt{(7 - 9)^2+(7 - 1)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10}$
For side $GJ$:
$x_1=-2,y_1=-3,x_2=7,y_2 = 7$
$GJ=\sqrt{(7 + 2)^2+(7 + 3)^2}=\sqrt{81+100}=\sqrt{181}$
Since all side - lengths are different, the triangle is scalene.

Answer:

a) Scalene
b) Scalene