Question

by looking at the graph it seems that the points (-5, -3), (-2, -4), and (5,1) do not form a right triangle. note that the right triangle has two sides perpendicular to each other. if (-5, -3), (-2, -4), and (5,1) are the vertices of a right triang
a. two sides of the triangle must have the sum of their slopes as 0.
b. two sides of the triangle must have the same slope.
c. two sides of the triangle must have the product of their slopes as -1.
d. two sides of the triangle must have the product of their slopes as 1.
what is the formula for the slope of a line through (x₁,y₁) and (x₂,y₂) with x₁≠x₂?
a. \\(\frac{y_2 - y_1}{x_2 - x_1}\\)
b. \\(\frac{y_2 - x_2}{y_1 - x_1}\\)
c. \\(\frac{x_2 - x_1}{y_2 - y_1}\\)
d. \\(\frac{y_1 - x_1}{y_2 - x_2}\\)
recall that the three points (-5, -3), (-2, -4), and (5,1) are labeled as a, b, and c respectively. find the slopes of each side of the triangle.
slope ( m_{ab} ) of the side ab = ( -\frac{1}{3} ) (type an integer or a simplified fraction.)
slope ( m_{bc} ) of the side bc = ( \frac{5}{7} ) (type an integer or a simplified fraction.)
slope ( m_{ac} ) of the side ac = \\( \square \\) (type an integer or a simplified fraction.)

Explanation:

Step1: Identify the formula for slope

The formula for the slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) (where \( x_1
eq x_2 \)) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identify the coordinates for points A and C

Point \( A \) is \((-5, -3)\) and point \( C \) is \((5, 1)\). So, \( x_1=-5 \), \( y_1 = - 3 \), \( x_2=5 \), \( y_2=1 \).

Step3: Apply the slope formula to points A and C

Substitute the values into the slope formula:
\( m_{AC}=\frac{y_2 - y_1}{x_2 - x_1}=\frac{1-(-3)}{5-(-5)} \)

Step4: Simplify the numerator and denominator

Simplify the numerator: \( 1 - (-3)=1 + 3 = 4 \)
Simplify the denominator: \( 5-(-5)=5 + 5 = 10 \)
So, \( m_{AC}=\frac{4}{10} \)

Step5: Simplify the fraction

Simplify \( \frac{4}{10} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
\( \frac{4\div2}{10\div2}=\frac{2}{5} \)

Answer:

\(\frac{2}{5}\)