Explanation:
Step 1. We are given the coordinates of the three endpoints of a triangle
R(-43,-4), S(0,-3), and T(13,-44)
Which are shown in the following diagram (not to scale):
Step 2. The problem states that Q is the midpoint of the line that goes from R to T (RT):
And we need to find the equation of the line QS, shown here in yellow:
Step 3. To find the equation of the line, first, we need to find the coordinates of Q. Since Q is the midpoint between R and T, we find its coordinates by averaging the coordinates of R and T:
Step 4. Then, once we know the coordinates of point Q, we need to find the slope between Q and S. To make it easier to find the slope, we will label the coordinates of S and Q as follows:
And use the slope formula:
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \downarrow \\ m=\frac{-24-(-3)}{-15-0} \\ \downarrow \end{gathered}[/tex][tex]\begin{gathered} m=\frac{-24+3}{-15} \\ \downarrow \\ m=\frac{-21}{-15} \\ \downarrow \\ m=\frac{7}{5} \end{gathered}[/tex]
The slope is m=7/5
Step 5. The final step is to use the point-slope formula using point S as (x1,y1) and the previously found slope m
[tex]\begin{gathered} point\text{ - slope formula:} \\ y-y_1=m(x-x_1) \end{gathered}[/tex]
Substituting the known values:
[tex]y-(-3)=\frac{7}{5}(x-0)[/tex]
Solving the operations and solving for y:
[tex]\begin{gathered} y+3=\frac{7}{5}x \\ \downarrow \\ y=\frac{7}{5}x-3 \end{gathered}[/tex]
Where the slope is m=7/5, and the y-intercept 'b' is b=-3.
Answer:
[tex]\begin{gathered} y=\frac{7}{5}x-3 \\ where \\ m=\frac{7}{5} \\ b=-3 \end{gathered}[/tex]
