Question

1. determine the slopes of each side length of a triangle if the vertices are a(4, -2), b(-4, 8), c(4, 8). bonus: can you determine what kind of triangle this is by thinking about the slopes?

Explanation:

Step1: Recall the slope formula

The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Find slope of \( AB \)

For points \( A(4, - 2) \) and \( B(-4, 8) \), substitute into the slope formula:
\( m_{AB}=\frac{8 - (-2)}{-4 - 4}=\frac{8 + 2}{-8}=\frac{10}{-8}=-\frac{5}{4} \)

Step3: Find slope of \( BC \)

For points \( B(-4, 8) \) and \( C(4, 8) \), substitute into the slope formula:
\( m_{BC}=\frac{8 - 8}{4 - (-4)}=\frac{0}{8}=0 \)

Step4: Find slope of \( AC \)

For points \( A(4, - 2) \) and \( C(4, 8) \), substitute into the slope formula:
\( m_{AC}=\frac{8 - (-2)}{4 - 4}=\frac{10}{0} \), which is undefined (vertical line).

Step5: Analyze the triangle type (Bonus)

• The slope of \( BC \) is \( 0 \), so \( BC \) is a horizontal line.
• The slope of \( AC \) is undefined, so \( AC \) is a vertical line.
• A horizontal line and a vertical line are perpendicular (since the product of their slopes is \( 0\times\text{undefined} \) but geometrically, horizontal and vertical lines are perpendicular). Also, we can check the lengths or the angles. Since \( AC \) is vertical (\( x = 4 \)) and \( BC \) is horizontal (\( y = 8 \)), they intersect at \( C(4,8) \) at a right angle. So the triangle is a right triangle.

Answer:

Slopes: \( m_{AB}=-\frac{5}{4} \), \( m_{BC} = 0 \), \( m_{AC} \) is undefined.
Bonus: The triangle is a right triangle (since \( AC \) is vertical and \( BC \) is horizontal, they are perpendicular, forming a right angle at \( C \)).