Question

If polygon ABCD is reflected over line f, where would point B’ be graphed?

Answer 1

(-2, -10)

EXPLANATION :

We need to get first the equation of the reflection line f.

The intercepts are (-3, 0) and (0, -3)

Using two point formula :

`\begin{gathered} y-y_1=(y_2-y_1)/(x_2-x_1)(x-x_1) \\ \\ y-0=(-3-0)/(0-(-3))(x+3) \\ \\ y=-(x+3) \\ y=-x-3 \end{gathered}`

The slope of the line is -1

We need to take the negative reciprocal of the slope, it will be the slope of the perpendicular line to line f.

So the slope will be 1.

Now, we need to find a line that has a slope of 1 and passes through point B.

The slope is 1, and it passes through point (7, -1)

Using point slope formula :

`\begin{gathered} y-y_1=m(x-x_1) \\ y+1=1(x-7) \\ y+1=x-7 \\ y=x-8 \end{gathered}`

So now we have two equations of the line :

`\begin{gathered} y=-x-3 \\ y=x-8 \end{gathered}`

Find the intersection of these two lines by equating y =y :

`\begin{gathered} y=y \\ -x-3=x-8 \\ -x-x=-8+3 \\ -2x=-5 \\ x=(-5)/(-2) \\ \\ x=2.5 \end{gathered}`

Substitute x = 2.5 to the first equation :

`\begin{gathered} y=-x-3 \\ y=-2.5-3 \\ y=-5.5 \end{gathered}`

So the intersection point is (2.5, -5.5)

Note that this intersection point is the midpoint of point B and point B'

The midpoint formula is :

`M((x_1+x_2)/(2),(y_1+y_2)/(2))`

So we have B(7, -1) and M(2.5, -5.5)

and we are going to solve for the coordinates of B'(x2, y2)

That will be :

`\begin{gathered} (2.5,-5.5)=((7+x_2)/(2),(-1+y_2)/(2)) \\ \\ 2.5=(7+x_2)/(2) \\ 5=7+x_2 \\ x_2=5-7 \\ x_2=-2 \\ \\ \\ -5.5=(-1+y_2)/(2) \\ -11=-1+y_2 \\ y_2=1-11 \\ y_2=-10 \end{gathered}`

Therefore, the coordinate of B' is (-2, -10)