Question

the points a(0,5), b(4,2), and c(0,2) form the vertices of a right - triangle in the coordinate plane. what is the equation of the line that forms the hypotenuse? the equation of the line is □. (type your answer in slope - intercept form.)

Explanation:

Step1: Calculate the slope

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $A(0,5)$ and $B(4,2)$, we have $x_1 = 0,y_1=5,x_2 = 4,y_2 = 2$. Then $m=\frac{2 - 5}{4 - 0}=\frac{-3}{4}=-\frac{3}{4}$.

Step2: Find the y - intercept

The slope - intercept form of a line is $y=mx + b$, where $b$ is the y - intercept. We know the line passes through the point $A(0,5)$. When $x = 0,y = 5$ in the equation $y=mx + b$. Substituting $x = 0,y = 5$ and $m=-\frac{3}{4}$ into $y=mx + b$, we get $5=-\frac{3}{4}\times0 + b$, so $b = 5$.

Step3: Write the equation of the line

Substitute $m = -\frac{3}{4}$ and $b = 5$ into the slope - intercept form $y=mx + b$. The equation of the line is $y=-\frac{3}{4}x+5$.

Answer:

$y = -\frac{3}{4}x + 5$