Answer:
B
Step-by-step explanation:
We know that AB has endpoints at A(4, -3) and B(3, 7).
To find Line D, which is the perpendicular bisector of AB, we first need to find the slope of AB and its midpoint.
So, let's find the slope of AB using the slope formula:
[tex]\displaystyle m=\frac{y_2 - y_1}{x_2 - x_1}[/tex]
Let (4, -3) be (x₁, y₁) and let (3, 7) be (x₂, y₂). Substitute them into the slope formula:
[tex]\displaystyle m=\frac{7-(-3)}{3-4}=\frac{10}{-1}=-10[/tex]
So, the slope of AB is -10.
Remember that the slopes of perpendicular lines are negative reciprocals of each other.
Therefore, the slope of our Line D is the negative reciprocal of -10. Which is -10 flipped and multiplied by a negative.
So, Line D has a slope of 1/10.
Now, since Line D is also the perpendicular bisector of AB, we need to find the midpoint of AB since Line D must pass through this point.
To find the midpoint, we can use the midpoint formula:
[tex]\displaystyle M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)[/tex]
We can again let (4, -3) be (x₁, y₁) and let (3, 7) be (x₂, y₂). Substitute them into the midpoint formula to obtain:
[tex]\displaystyle M=\left(\frac{4+3}{2},\frac{-3+7}{2}\right)[/tex]
Evaluate. So, the midpoint is:
[tex]\displaystyle M=\left(\frac{7}{2},2\right)=(3.5,2)[/tex]
Now, we can find the equation of our Line D. We know that its slope is 1/10, and that it must pass through (3.5, 2). So, we can use the point-slope form:
[tex]y-y_1=m(x-x_1)[/tex]
Where m is the slope. Let's let (3.5, 2) be (x₁, y₁) and substitute 1/10 for m. This yields:
[tex]\displaystyle y-2=\frac{1}{10}(x-3.5)[/tex]
This is the same as choice C. So, choice C is indeed an equation of Line D.
We can distribute the right to obtain:
[tex]\displaystyle y-2=\frac{1}{10}x-0.35[/tex]
Add 2 to both sides to acquire:
[tex]\displaystyle y=\frac{1}{10}x+1.65[/tex]
This is the same as choice D. So, choice D is indeed an equation of Line D.
Moreover, we can put this into standard form by multiply everything by 10, which gives:
[tex]10y=x+16.5[/tex]
Subtracting x from both sides yields:
[tex]-x+10y=16.5[/tex]
This is the same as A. So, choice A is also an equation of Line D.
There is no way to acquire choice B. So, B is not an equation of Line D.
Therefore, our answer is B.
And we're done!
