Question

Please solve this problem with full sentences and explain it please.

Question 2 
(02.05 MC)

Triangle PQR is transformed to triangle P'Q'R'. Triangle PQR has vertices P(8, 0), Q(6, 2), and R(−2, −4). Triangle P'Q'R' has vertices P'(4, 0), Q'(3, 1), and R'(−1, −2).

Plot triangles PQR and P'Q'R' on your own coordinate grid.

Part A: What is the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R'? Explain your answer. (4 points)

Part B: Write the coordinates of triangle P"Q"R" obtained after P'Q'R' is reflected about the y-axis. (4 points)

Part C: Are the two triangles PQR and P''Q''R'' congruent? Explain your answer. (2 points)

Answer 1

ANSWER TO QUESTION 1.

We plot the points:

`P(8,0)`

`Q(6,2)`

and

`R(-2,-4)`

We plot the image triangle also with the points,

`P'(4,0)`

`Q'(3,1)`

and

`R(-1,-2)`

See graph in attachment.

ANSWER TO QUESTION 2.

The scale factor is the ratio of the image length to the object length.

`Scale\: factor=(|P'Q'|)/(|PQ|)`.

We need to use the distance formula to determine the two lengths.

`d=√((x_2-x_1)^2+(y_2-y_1)^2)`

`|P'Q'|=√((3-4)^2+(1-0)^2)`

`|P'Q'|=√((-1)^2+(1)^2)`

`|P'Q'|=√(1+1)`

`|P'Q'|=√(2)`

`|PQ|=√((6-8)^2+(2-0)^2)`

`|PQ|=√((-2)^2+(2)^2)`

`|PQ|=√(4+4)`

`|PQ|=√(8)`

`|PQ|=2√(2)`

Therefore the scale factor of the dilation is

`Scale\: factor=(√(2))/(2√(2))`.

`Scale\: factor=(1)/(2)`.

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:

`P'(4,0)\rightarrow P''(-4,0)`

`Q'(3,1)\rightarrow Q''(-3,1)`

and

`R(-1,-2)\rightarrow R''(1,-2)`

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.