Lesson 3
Dilations with no Grid
Let’s dilate figures not on grids.
Problem 1
Segment \(AB\) measures 3 cm. Point \(O\) is the center of dilation. How long is the image of \(AB\) after a dilation with . . .
- Scale factor 5?
- Scale factor 3.7?
- Scale factor \(\frac 1 5\)?
- Scale factor \(s\)?
Problem 2
Here are points \(A\) and \(B\). Plot the points for each dilation described.
![Points A and B](https://cms-im.s3.amazonaws.com/kWYCmiQXpZ9nAnADWXPXqhZc?response-content-disposition=inline%3B%20filename%3D%228-8.2.A2.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.A2.Image.10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235823Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0e67c6343aed5e84abfc85804367e1a875a4bf69680bc6c7642bae104c77c7c7)
- \(C\) is the image of \(B\) using \(A\) as the center of dilation and a scale factor of 2.
- \(D\) is the image of \(A\) using \(B\) as the center of dilation and a scale factor of 2.
- \(E\) is the image of \(B\) using \(A\) as the center of dilation and a scale factor of \(\frac 1 2\).
- \(F\) is the image of \(A\) using \(B\) as the center of dilation and a scale factor of \(\frac 1 2\).
Problem 3
Make a perspective drawing. Include in your work the center of dilation, the shape you dilate, and the scale factor you use.
Problem 4
Triangle \(ABC\) is a scaled copy of triangle \(DEF\). Side \(AB\) measures 12 cm and is the longest side of \(ABC\). Side \(DE\) measures 8 cm and is the longest side of \(DEF\).
- Triangle \(ABC\) is a scaled copy of triangle \(DEF\) with what scale factor?
- Triangle \(DEF\) is a scaled copy of triangle \(ABC\) with what scale factor?
Problem 5
The diagram shows two intersecting lines.
Find the missing angle measures.
![Two intersecting lines, forming an X.](https://cms-im.s3.amazonaws.com/H8WVUfQkpBy1WX9Z7aSBCXP2?response-content-disposition=inline%3B%20filename%3D%228-8.1.D14.newPP.01b.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.D14.newPP.01b.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235823Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=520139588c5405cb045b4894a6ef410d3f72d095ebd96fdd497a9fa13a1f505a)
Problem 6
- Show that the two triangles are congruent.
- Find the side lengths of \(DEF\) and the angle measures of \(ABC\).
![Two triangles in an x y plane.](https://cms-im.s3.amazonaws.com/dMAZJj8rqoo3ezNig9m6dFgD?response-content-disposition=inline%3B%20filename%3D%228-8.1.C.PP.Image.07b.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.C.PP.Image.07b.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235823Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=fe13a8df6fbf179074af46d10025c5e9350e3d2a900b8f30d0b336c5edcd0f1a)