Point A' 3 5 is the reflection of point A -1 3 What is the point of reflection?

Answer 1

(1,4)

Let (x,y) represent the point of reflection. We know that A and A' must be on the same line as (x,y) and must be equidistant from (x,y).

Because they are equidistant from (x,y), we know:

(x-3)2+(y-(5))2=(x-(-1))2+(y-(3))2

Now find the line that they are on by using the points A and A':

5=m(3)+b

3=m(-1)+b

Subtracting the left and right hand sides of the two equations, gives us:

2=4m -->m=1/2

Using this value of m and substituting into either equation gives us b=7/2.

So, all three points are on the line:

y=1/2*x+7/2

This is equivalent to x=2y-7. I will substitute this into the first equation to find our point [I could have used y=1/2*x+7/2, but I'm using the value of x to avoid the fractions.]

((2y-7)-3)2+(y-(5))2=((2y-7)-(-1))2+(y-(3))2

(2y-10)2+(y-5)2=(2y-6)2+(y-3)2

4y2-40y+100+y2-10y+25=4y2-24y+36+y2-6y+9

-50y+125=-30y+45

20y=80

y=4

Substituting y=4 into x=2y-7, gives x=1.