Question

1. If mapping $M:\left(x,y\right)\to \left(2x,2y\right)$, find the images of $P$ and $Q$ in the diagram.
2. Is $M$ a transformation?
3. Does $M$ appear to be an isometry?
4. Decide whether $M$ maps the midpoint of $\overline{PQ}$ to the midpoint of $\overline{P\text{'}Q\text{'}}$.

Short answer

1. The images of P and Q are $P\text{'}\left(2,6\right)$ and $Q\text{'}\left(8,2\right)$ respectively.
2. The mapping M is a transformation.
3. The mapping M does not appear to be an isometry.
4. The mapping M does map to the midpoint $\overline{PQ}$to the midpoint of$\overline{P\text{'}Q\text{'}}$.

Step-by-step solution

a.Step 1. Given Information.

The mapping is $M:\left(x,y\right)\to \left(2x,2y\right)$ and the diagram is:

Step 2. Calculation.

From the image, the points are $P\left(1,3\right)$ and $Q\left(4,1\right)$.

To find the images of P, substitute 1 for x and 3 for y in the mapping.

$\begin{array}{c}M:\left(1,3\right)\to \left(2\cdot 1,2\cdot 3\right)\\ M:\left(1,3\right)\to \left(2,6\right)\\ M:\left(1,3\right)\to P^{\text{'}}\end{array}$

To find the images of Q, substitute 4 for x and 1 for y in the mapping.

$\begin{array}{c}M:\left(4,1\right)\to \left(2\cdot 4,2\cdot 1\right)\\ M:\left(4,1\right)\to \left(8,2\right)\\ M:\left(4,1\right)\to Q^{\text{'}}\end{array}$

Step 3. Conclusion.

Therefore, the images of P and Q are $P\text{'}\left(2,6\right)$ and $Q\text{'}\left(8,2\right)$ respectively.

b.Step 1. Given Information.

The mapping is $M:\left(x,y\right)\to \left(2x,2y\right)$.

Step 2. Explanation.

Transformation: A one-to-one mapping from the whole plane to the whole plane.

The given case is also one of the examples of one-to-one mapping because all the pre-image have their images and all the images have their pre-images.

Step 3. Conclusion.

Therefore, the mapping M is a transformation.

c.Step 1. Given Information.

The mapping is $M:\left(x,y\right)\to \left(2x,2y\right)$ and the diagram is:

Step 2. Calculation.

From the image, the points are $P\left(1,3\right)$and $Q\left(4,1\right)$. Form part (a), the images of P and Q are $P\text{'}\left(2,6\right)$ and $Q\text{'}\left(8,2\right)$ respectively.

The formula for the distance between two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by,

$\text{distance}=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

To find the length of PQ, substitute 4 for $x_2$, 1 for $x_1$, 1 for $y_2$, 3 for $y_1$ in the above formula.

$\begin{array}{c}PQ=\sqrt{{\left(4-1\right)}^2+{\left(1-3\right)}^2}\\ =\sqrt{{\left(3\right)}^2+{\left(-2\right)}^2}\\ =\sqrt{9+4}\\ =\sqrt{13}\end{array}$

To find the length of $P\text{'}Q\text{'}$, substitute 8 for $x_2$, 2 for $x_1$, 2 for $y_2$, 6 for $y_1$ in the above formula.

$\begin{array}{c}{P}^{\text{'}}{Q}^{\text{'}}=\sqrt{{\left(8-2\right)}^2+{\left(2-6\right)}^2}\\ =\sqrt{{\left(6\right)}^2+{\left(-4\right)}^2}\\ =\sqrt{36+16}\\ =\sqrt{52}\end{array}$

The given mapping is not an isometry because $\overline{PQ}\ne \overline{P\text{'}Q\text{'}}$.

Step 3. Conclusion.

Therefore, the mapping M does not appear to be an isometry.

d.Step 1. Given Information.

The mapping is $M:\left(x,y\right)\to \left(2x,2y\right)$.

Step 2. Calculation.

Let N is the midpoint of the line PQ and $N\text{'}$is the midpoint of the line$P\text{'}Q\text{'}$. The points are $P\left(1,3\right)$and $Q\left(4,1\right)$. The images of points are $P\text{'}\left(2,6\right)$and $Q\text{'}\left(8,2\right)$respectively.

The point N can be calculated as:

$\begin{array}{c}N=\left(\frac{1+4}{2},\frac{3+1}{2}\right)\\ =\left(\frac{5}{2},2\right)\end{array}$

The point $N\text{'}$can be calculated as:

$\begin{array}{c}{N}^{\text{'}}=\left(\frac{2+8}{2},\frac{6+2}{2}\right)\\ =\left(5,4\right)\end{array}$

Now, from the mapping image of N,

$\begin{array}{c}M:\left(\frac{5}{2},2\right)\to \left(2\cdot \frac{5}{2},2\cdot 2\right)\\ M:\left(\frac{5}{2},2\right)\to \left(5,4\right)\\ M:\left(\frac{5}{2},2\right)\to N^{\text{'}}\end{array}$

Step 3. Conclusion.

Therefore, the mapping M does map to the midpoint $\overline{PQ}$ to the midpoint of $\overline{P\text{'}Q\text{'}}$.