10.3: Transformations that Change Size and Similar Figures

This section covers transformations that either enlarge or shrink an object from a point, P. The point P is called the center of the size transformation. The multiplier used to enlarge or shrink the object is called the scale factor, k. The calculations used to make the transformation depend on the distances from point P to the vertices of the object. These distances are multiplied by k. So, to enlarge or shrink an object, find the distance from the point P to a vertex A of the object. Multiply this distance by k to get kPA, where PA represents the distance from P to A. Then, measure this new distance, kPA, from point P in the direction of vertex A. This distance gives the new location of vertex A after the object has been size-transformed. Repeat for all vertices of the object.

Example \(\PageIndex{1}\): Enlarge a Triangle by a Factor of Two

Figure \(\PageIndex{1}\): Triangle to be Size-Transformed by a Factor of Two

	B
	P				A

Step 1: Measure the distances from point P to each vertex of the triangle. One vertex of the triangle is on point P, so that vertex will remain at point P.

The distance from point P to vertex A is three units.

The distance from point P to vertex B is also three units.

Step 2: Multiply these distance by the scale factor two.

2PA = 2(3) = 6

2PB = 2(3) = 6

Step 3: Measure six units from point P in the direction of vertex A and measure six units from point P in the direction of vertex B. The new locations of A and B are each six units from P in their corresponding directions as shown below.

Figure \(\PageIndex{2}\): Triangle Enlarged by a Factor of Two

	B’
	B
	P				A			A’

Example \(\PageIndex{2}\): Enlarge a Diamond by a Factor of Two

Figure \(\PageIndex{3}\): Diamond to be Size-Transformed by a Factor of Two

					B
		A					D
					C
	P

Step 1: Measure the distances from point P to each vertex of the diamond.

The distance from point P to vertex A is PA

Likewise, the distances from point P to the other three vertices are PB, PC, and PD, respectively.

Step 2: Multiply these distance by the scale factor two.

The distances of the new points from P are: 2PA, 2PB, 2PC, and 2PD.

Step 3: Measure these distances from point P in the direction of each vertex A, B, C, and D as shown below.

Figure \(\PageIndex{4}\): Diamond Enlarged by a Factor of Two

							B'
				A'						D'
				B
		A				D
							C'
				C
	P

Example \(\PageIndex{3}\): Shrink a Trapezoid by a Factor of

Figure \(\PageIndex{5}\): Trapezoid to be Size-Transformed by a Factor of

		B						C
						P
	A									D

Step 1: Measure the distances from point P to each vertex of the trapezoid.

The distances from point P to the vertices are PA, PB, PC, and PD, respectively.

Step 2: Multiply these distances by the scale factor .

The distances of the new points from P are: PA, PB, PC, and PD.

Step 3: Measure these distances from point P in the direction of each vertex A, B, C, and D as shown below.

Figure \(\PageIndex{6}\): Trapezoid Shrunk by a Factor of

		B						C
				B'
								C'
						P
			A'				D'
	A									D

Shapes that have been transformed by a enlarging or shrinking are similar figures to the original shape.

Similarity Using Transformations

Two figures are similar if and only if there exists a combination of an isometry (rigid motion) and a size transformation that generates one figure as the image of the other.

Similar figures are figures that have the same shape but not necessarily the same size. Side lengths and interior angles of similar figures are proportional to each other.

Figure \(\PageIndex{7}\): Similar Figures

The two rectangles below are similar if the sides are proportional to each other. In other words, they are similar if .

a c

b d
The two rectangles are related by the scale factor k. Therefore, the sides of the rectangles are related to each other by: .

Let = perimeter of the smaller rectangle and = perimeter of the larger rectangle.

, but remember that , so

, now factor out the to get

, and also, , so

If represents the perimeter of the smaller rectangle and represents the perimeter of the larger rectangle, then the two perimeters are related by .

Let = area of the smaller rectangle and = the area of the larger rectangle.

, but remember that , so

, rearrange to get

, and also, , so

If represents the area of the smaller rectangle and represents the area of the larger rectangle, then the two areas are related by .

Example \(\PageIndex{4}\): Areas and Perimeters of Similar Triangles

Figure \(\PageIndex{8}\): The Triangles in this Figure are Similar Triangles

4 16

a. Determine the perimeter of the larger triangle if the perimeter of the smaller triangle is 12 mm.

b. Determine the area of the larger triangle if the area of the smaller triangle is 6.9 mm2

First, find the scale factor using the fact that similar triangles have sides that are proportional to each other: .

1. 

2. 

Gnomons

A gnomon is a shape which, when added to a shape , yields another shape that is similar to the original shape . Alternately, a gnomon is also the shape which, when subtracted from a shape , yields another shape which is similar to the original shape .

Figure \(\PageIndex{9}\): What is a Gnomon?

The blue rectangle shown below is the original shape . When the red L-shape is attached to the blue rectangle, a new rectangle is formed, called shape . If the red L-shape is a gnomon to shape , then the blue rectangle (shape ) is similar to the blue and red rectangle (shape ). Also, since the rectangles are similar, the side lengths are proportional:

b b + y

a a + x

Example \(\PageIndex{5}\): Rectangular Gnomon

Find the value of x so that the red larger rectangle on the right is a gnomon to the blue smaller rectangle on the left.

Figure \(\PageIndex{10}\): Rectangular Gnomon

3 x

9



To calculate this, set up a proportion so that the sides of the small blue rectangle on the left are proportional to the sides of the blue and red rectangles (left and right) combined. To set up the proportion, make ratios of the width over the length of the small blue rectangle on the left and the width over the length of the combined rectangles.

If , then the red larger rectangle is a gnomon to the blue smaller rectangle.

Example \(\PageIndex{6}\): Triangular Gnomon

Find the values of x and y so that the larger red triangle on the left is a gnomon to the smaller blue triangle on the right.

Figure \(\PageIndex{11}\): Triangular Gnomon

x 15

y 9

To calculate this, set up proportions, one with x and one with y, so that the sides of the smaller blue triangle on the right are proportional to the sides of the blue and red triangles combined. To set up the proportion for x, make ratios of the longer leg over the shorter leg of the smaller blue triangle on the right and the longer leg over the shorter leg of the combined triangles.

Now to set up the proportion for y, make ratios of the shorter leg over the hypotenuse of the smaller blue triangle on the right and the shorter leg over the hypotenuse of the combined triangles.

If and , then the red larger triangle on the left is a gnomon to the blue smaller triangle on the right.