Similarity Transformations

Given $∆ABC$ below with coordinates$A(2,1),B(2,3),C(5,1)$, a center of dilation $(0,0)$, and a scale factor of 2.

Apply the dilation to the polygon and plot $∆A\text{'}B\text{'}C\text{'}$.

Solution:

Step 1: We will start by drawing the pre-image $∆ABC$.

Step 2: From the question, we see that the scaling factor is 2. Notice that the scale factor is greater than 1. This means that it's an enlargement. So, you multiply each x and y co-ordinate of $∆ABC$ by the scaling factor, 2. See the table below.

The figures on the graph below show shapes with coordinates $P(2,2),Q(2,4),R(6,4),S(6,2)andW(5,5),X(5,9),Y(12,9),Z(12,5).$ Determine if they are similar or not.

Step 1: Make a diagram

As you can see from the graph, the figures have been dilated meaning that some transformation has taken place.

Step 2: The fastest way to check their similarities is to compare the coordinates of both figures. The first figure is $PQRS$and the second is $WXYZ$. First, we need to identify corresponding vertices. The rectangles are both in landscape orientation, so we can match the bottom left on one to the bottom left of the other and so on. You should get that P corresponds to W (both bottom left), S corresponds to Z (bottom right), Q to X (top left), and R to Y (top right).

Next, we check

$\frac{Px}{Wx}=\frac{Py}{Wy}=\frac{Sx}{Sy}=\frac{Zx}{Zy}=\frac{Qx}{Xx}=\frac{Qy}{Xy}=\frac{Rx}{Yy}$

Step 3: Subbing in coordinates in the above equation, we get

$\frac{2}{5}=\frac{2}{5}\ne \frac{6}{12}\ne \frac{2}{5}=\frac{2}{5}\ne \frac{4}{9}\ne \frac{6}{12}\ne \frac{4}{9}$

As you can see, the coordinates are not equal so similarity does not exist.

You'll get the ratio of each coordinate to see if they will have the same scale factor.

The first set of coordinates to compare is $P(2,2)andW(5,5)$. The ratio will be $\frac{2}{5}:\frac{2}{5}$. They are equal.

The second set is $Q(2,4)andX(5,9).$ (The ratio will be $\frac{2}{5}:\frac{4}{9}$. Our result is not equal meaning that the figures cannot be similar. If we had got $\frac{2}{5}$ instead of $\frac{4}{9}$, then we would have continued with the next coordinate to see if we would get another $\frac{2}{5}$ as the scale factor. The scale factor differs so, the figures are not similar.

Another way to go about this is to look at the graph itself. You should compare the corresponding sides of the figures and see if they are of the same proportion.

From the figure, $\overline{PQ}$ is 2 and $\overline{WX}$ is 4. The proportion here is $\frac{2}{4}$ which is $\frac{1}{2}$.

$\overline{QR}$ is 4 and $\overline{XY}$ is 7. The proportion here is $\frac{4}{7}$. $\frac{1}{2}$ is very different from $\frac{4}{7}$.

Therefore, similarity does not exist.

Determine if the following figures are similar or not.

You can't really tell if they are similar by just looking at it because they are placed in different positions. Let's rotate and then translate the first figure and see the outcome.

After rotating and translating, the above figure was produced. As you can see, they are both in the same orientation and you can now make comparisons using corresponding parts.

$\frac{30}{40}=\frac{21}{28}=\frac{3}{4}$

The parts are in the same proportion. Therefore, they are similar.