Unit 7 Toolbox

A dilation is a non-rigid transformation that enlarges or reduces a geometric figure by a scale factor relative to a point.  The scale factor is usually labeled k.  

Refer to each chapter in the GeoGebra Similarity Book to answer the following questions.

Scale Change Exploration 
https://www.geogebra.org/m/d6xuhjws#material/bacj7nye

1

Which word describes the type of change if k > 1?

1

Which word describes the type of change if 0 < k < 1?

1

Which word describes the type of change if k = 1?

You may have noticed that multiplying each x and y coordinate in the pre-image would generate the corresponding points in the image.  

We write this pattern like this:  (x , y) ----> (kx , ky) 
(x , y) is the pre-image point and (kx , ky) is the image point under a dilation.

Proving Similarity with Transformations
https://www.geogebra.org/m/d6xuhjws#chapter/512777

Two figures are similar figures if and only if the pre-image figure can be mapped onto the image figure under one transformation or a series of transformations.  

All similar figures are the same shape because angle measure, betweeness, collinearity is preserved (unchanged).  Distance is only preserved in similar figures that are also congruent.  This means that dilations that don't have a k value of 1 are non-rigid.

The notation for the similar triangles in Transformation #1 is written below where corresponding parts are in the same position in the triangle name similar to the way congruence statements are written. The pre-image is listed first in the similarity statement.

Properties of Scale Change
https://www.geogebra.org/m/d6xuhjws#chapter/569501

Refer to Properties of Scale Change #1 when answering questions #4 and #5.

1

Every dilation or scale change has a center. Where is the center of every scale change?

1

The pre-image, image points and the center of the dilation are all collinear (on the same line).

Refer to Properties of Scale Change #2 when answering questions #6 and #7.

1

Under a scale change, corresponding angle pairs are. . .

1

Under a scale change, corresponding side lengths are always. . .

Notice that the ratio of image to pre-image lengths is always equal to k (the size change factor).

Refer to Properties of Scale Change #3 when answering questions #8 and #9.

1

Under a scale change, any pair of corresponding pre-image sides and images sides are parallel.

1

Under a scale change, the ratio of any corresponding pair of image sides lengths and pre-image side lengths will NOT always equal the scale factor.

Refer to Perimeter and Area to answer questions #10-14.

1

The perimeter ratio and area ratio are always equal.

1

The perimeter ratio is equal to the scale factor.

1

The area ratio is equal to the scale factor.

1

The area ratio is equal to the square of the scale factor.

1

Based on these observations, it seems that the volume ratio will be equal to the _______________ of the scale factor.

Triangle Similarity Conjectures
https://www.geogebra.org/m/d6xuhjws#chapter/569519

1

Refer to Definition of Triangle Similarity to answer this question. Which of these are true about similar triangles.

1

Refer to Triangle Similarity Conjectures when answering this question. Which of these can be used to determine similar triangles?

1

When working with Triangle Similarity Conjectures (such as SAS), each A represents ____________.

1

When working with Triangle Similarity Conjectures (such as SAS), each S represents ____________.

Side-Splitting Conjecture
https://www.geogebra.org/m/d59vydwz

1

Use the drawing in the "show your work" section to choose a true proportion using a, b, c and d based on Triangle Side-Splitting.

1

Unit 7 Toolbox

Which word describes the type of change if k > 1?

Which word describes the type of change if 0 < k < 1?

Which word describes the type of change if k = 1?

Every dilation or scale change has a center. Where is the center of every scale change?

The pre-image, image points and the center of the dilation are all collinear (on the same line).

Under a scale change, corresponding angle pairs are. . .

Under a scale change, corresponding side lengths are always. . .

Under a scale change, any pair of corresponding pre-image sides and images sides are parallel.

Under a scale change, the ratio of any corresponding pair of image sides lengths and pre-image side lengths will NOT always equal the scale factor.

The perimeter ratio and area ratio are always equal.

The perimeter ratio is equal to the scale factor.

The area ratio is equal to the scale factor.

The area ratio is equal to the square of the scale factor.

Based on these observations, it seems that the volume ratio will be equal to the _______________ of the scale factor.

Refer to Definition of Triangle Similarity to answer this question. Which of these are true about similar triangles.

Refer to Triangle Similarity Conjectures when answering this question. Which of these can be used to determine similar triangles?

When working with Triangle Similarity Conjectures (such as SAS), each A represents ____________.

When working with Triangle Similarity Conjectures (such as SAS), each S represents ____________.

Use the drawing in the "show your work" section to choose a true proportion using a, b, c and d based on Triangle Side-Splitting.

Use the drawing in the "show your work" section to choose a true proportion using a, b, c and d based on Trapezoid Side-Splitting.

Which word describes the type of change if k > 1?

Which word describes the type of change if 0 < k < 1?

Which word describes the type of change if k = 1?

Every dilation or scale change has a center. Where is the center of every scale change?

The pre-image, image points and the center of the dilation are all collinear (on the same line).

Under a scale change, corresponding angle pairs are. . .

Under a scale change, corresponding side lengths are always. . .

Under a scale change, any pair of corresponding pre-image sides and images sides are parallel.

Under a scale change, the ratio of any corresponding pair of image sides lengths and pre-image side lengths will NOT always equal the scale factor.

The perimeter ratio and area ratio are always equal.

The perimeter ratio is equal to the scale factor.

The area ratio is equal to the scale factor.

The area ratio is equal to the square of the scale factor.

Based on these observations, it seems that the volume ratio will be equal to the _______________ of the scale factor.

Refer to Definition of Triangle Similarity to answer this question. Which of these are true about similar triangles.

Refer to Triangle Similarity Conjectures when answering this question. Which of these can be used to determine similar triangles?

When working with Triangle Similarity Conjectures (such as SAS), each A represents ____________.

When working with Triangle Similarity Conjectures (such as SAS), each S represents ____________.

Use the drawing in the "show your work" section to choose a true proportion using a, b, c and d based on Triangle Side-Splitting.

Use the drawing in the "show your work" section to choose a true proportion using a, b, c and d based on Trapezoid Side-Splitting.