Question

What is the slope-intercept equation that represents a midsegment of the triangle below?
*Hint* There are three possible answers.

Answer 1

Answer:

1) $y=-6x-6.5$

2) $y=0.6x+0.1$

3) $y=-0.5x+1.75$

Step-by-step explanation:

Since we know that the segment joining the mid-points of two sides of a triangle is known as mid-segment of triangle. A triangle has three mid-segments.

1) Let us find midpoints of side ED and DF using midpoint formula.

$\text{Midpoint}=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$\text{Midpoint of segment DE}=(\frac{-4+1}{2},\frac{1+4}{2})$

$\text{Midpoint of segment DE}=(\frac{-3}{2},\frac{5}{2})=(-1.5,2.5)$

$\text{Midpoint of segment DF}=(\frac{-4+2}{2},\frac{1-2}{2})$

$\text{Midpoint of segment DF}=(\frac{-2}{2},\frac{-1}{2})=(-1,-0.5)$

Now let us find slope of line joining points (-1.5,2.5) and (-1,-0.5) using slope formula.

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{-0.5-2.5}{-1--1.5}$

$m=\frac{-3}{-1+1.5}$  

$m=\frac{-3}{0.5}=-6$        

Now let us substitute our values in slope intercept form of the line to find y-intercept.

$y=mx+b$, where m= slope of the line, b= y-intercept.

$-0.5=-6(-1)+b$  

$-0.5=6+b$  

$-6.5=b$

Upon substituting m=-6 and b= -6.5 in slope intercept form of line, we will get,

$y=-6x-6.5$

Therefore, 1st equation that represents the mid-segment parallel to side EF of the given triangle will be $y=-6x-6.5$.  

2) Let us find midpoint of segment EF.

$\text{Midpoint of segment EF}=(\frac{1+2}{2},\frac{4-2}{2})$

$\text{Midpoint of segment EF}=(\frac{3}{2},\frac{2}{2})=(1.5,1)$

Now let us find the slope of the line passing through points (1.5,1) and (-1,-0.5).

$m=\frac{1--0.5}{1.5--1}$        

$m=\frac{1.5}{2.5}=0.6$

Now let us find y-intercept of line parallel to segment DE.

$-0.5=0.6(-1)+b$

$-0.5=-0.6+b$    

$0.1=b$

Upon substituting m=0.6 and b=0.1 in slope intercept form of line we will get,

$y=0.6x+0.1$

Therefore, second equation that represents the mid-segment parallel to side DE of the given triangle will be $y=0.6x+0.1$.

3) Now let us find the slope of the mid-segment joining mid-points of segment DE and EF that are (-1.5,2.5) and (1.5,1).    

$m=\frac{1-2.5}{1.5--1.5}$        

$m=\frac{-1.5}{3}=-0.5$

Now let us find y-intercept of our line.

$y=mx+b$  

$1=-0.5(1.5)+b$  

$1=-0.75+b$  

$1.75=b$      

$y=-0.5x+1.75$

Therefore, the third equation that represents the mid-segment parallel to side DF of the given triangle will be $y=-0.5x+1.75$.