The Similarities Between Shapes

I have three examples for you.

Determine similarities by using transformations

Example # 1

A transformation is an operation that maps a geometric figure, preimage, onto a new figure, image.

A reflection is a transformation where a figure is flipped over a line.

Step 1 - reflect < ABC that's on the left top corner over the Y axis. So that A maps onto A on the bottom right hand corner.

Step 2 - Write the ratios comparing the lengths of each side.
AB/AB = 2/2 or 1,  AC/AC = 2/2 or 1, CB/CB = 3/3 or 1

Since the ratios are equal, triangle ABC (on the left side) is the dilated image of triangle ABC (on the right side). So the two triangles are similar because a translation maps triangle ABC (on the left side) onto triangle ABC (on the right side).

Example # 2

The orientation of the figures is different, so one of the transformations is reflection.

Reflect the figure that's on top over the Y axis so that P matches with P at the bottom so that it is oriented the same way as triangle PRQ.

write ratios comparing the length of each side.
PQ/PQ = 2/2 or 1,  RQ/RQ = 3/3 or 1,  PR/PR

The ratios are equal so the triangles are similar.

Example #3

Emily enlarges the photo shown by a scale factor of 2 for her web page. She then enlarges the webpage photo by a scale factor of 1.5 to print. If the original picture is 2 inches by 3 inches, what is the the dimensions of the print? Are the enlarged photos similar to the original?

Multiply each dimension of the original photo by 2 to find the dimensions of the webpage photo.
2 in. x  2 = 4 in.         3 in. x  2 = 6 in.

So, the webpage photo with be 4 inches by 6 inches. Multiply the dimensions of that photo by 1.5 to find the dimensions of the print.
4 in. x 1.5 = 6 in.      6 in. x 1.5 = 9 in.

The printed photo will be 6 inches by 9 inches. All of the three photos are similar since each enlargement was the result of a dilation.

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