Graph to include relection and values

Question

On the graph the endpoints are A (0,2) and B(9,4).

 

I had to draw a reflection line and those points are A(0,-2) and B(9,-4).

 

1.  Help with locating a point C on the x-axis such that AC + BC is minimized.

 

C = ( ?, 0)

 

 

2.  Fill in the values that complete the reflections with matrix multiplication.

 

                            A           B                  A'         B'

_____  _____     _____    ______     =   _____  ______

 

_____  _____     _____    ______     =   _____  ______

 

thanks!

Answer 1

Given that we have a reflection over the x-axis and we are trying to C that attains the minimalist value. We must understand the shortest line in the Cartesian plain is a straight line, thus since C is on the x-axis we cannot use AC+BC to find the minimalist value. We can however use the fact that since this is a reflection over the x-axis and point C is on the x-axis we can say that the distance between [A-C]=[A'-C] (code for absolute value)

Thus AC+BC=A'C+BC. Thus for slope and use the slope attained to solve by slope-intercept.

m=(y_2-y_1)/(x_2-x_1)=(4-(-2))/(9-0)=(6/9)=2/3

Solve for x:

0=(2/3)x-2

0+2=(2/3)x-2+2

2=(2/3)x+0 (using the additive identity property)

2=(2/3)x

3*2=3*(2/3)x

6=1*2x (using the multiplicative identity property)

6=2x

6/2=(2/2)x

3=x

Thus x is 3. So the point C that attains the minimalist value is C=(3,0)

Now to show that matrix multiplication we use the following matrix [1,0;0,-1]. That will attain the results for A' and B'