ANSWER TO QUESTION 1.
We plot the points:
[tex]P(8,0)[/tex]
[tex]Q(6,2)[/tex]
and
[tex]R(-2,-4)[/tex]
We plot the image triangle also with the points,
[tex]P'(4,0)[/tex]
[tex]Q'(3,1)[/tex]
and
[tex]R(-1,-2)[/tex]
See graph in attachment.
ANSWER TO QUESTION 2.
The scale factor is the ratio of the image length to the object length.
[tex]Scale\: factor=\frac{|P'Q'|}{|PQ|}[/tex].
We need to use the distance formula to determine the two lengths.
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
[tex]|P'Q'|=\sqrt{(3-4)^2+(1-0)^2}[/tex]
[tex]|P'Q'|=\sqrt{(-1)^2+(1)^2}[/tex]
[tex]|P'Q'|=\sqrt{1+1}[/tex]
[tex]|P'Q'|=\sqrt{2}[/tex]
[tex]|PQ|=\sqrt{(6-8)^2+(2-0)^2}[/tex]
[tex]|PQ|=\sqrt{(-2)^2+(2)^2}[/tex]
[tex]|PQ|=\sqrt{4+4}[/tex]
[tex]|PQ|=\sqrt{8}[/tex]
[tex]|PQ|=2\sqrt{2}[/tex]
Therefore the scale factor of the dilation is
[tex]Scale\: factor=\frac{\sqrt{2}}{2\sqrt{2}}[/tex].
[tex]Scale\: factor=\frac{1}{2}[/tex].
ANSWER TO QUESTION 3
For a reflection in the y-axis we negate the x-coordinates. That means we are using the y-axis as our mirror line. The mapping is given as follows:
[tex]P'(4,0)\rightarrow P''(-4,0)[/tex]
[tex]Q'(3,1)\rightarrow Q''(-3,1)[/tex]
and
[tex]R(-1,-2)\rightarrow R''(1,-2)[/tex]
ANSWER TO QUESTION 4
We can see from the diagram that, PQR is bigger than P"Q"R".
The reason is that PQR was reduced by scale factor of half and then reflected in the y-axis to obtain P"Q"R".
Since dilation is not a rigid motion, the two triangles PQR and P''Q''R" are not equal in all respect, therefore they are not congruent.
