Lesson 2
Circular Grid
Let’s dilate figures on circular grids.
Problem 1
Here are Circles \(c\) and \(d\). Point \(O\) is the center of dilation, and the dilation takes Circle \(c\) to Circle \(d\).
![Dilation](https://cms-im.s3.amazonaws.com/U51Yr1kzgPYwUutCrmpth1Az?response-content-disposition=inline%3B%20filename%3D%228-8.2.A2.newPP.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.A2.newPP.Image.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001034Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a53ec11899823b5da37ae055e4baff9a848504ace18cab8ad62ca4fe8038f3e2)
- Plot a point on Circle \(c\). Label the point \(P\). Plot where \(P\) goes when the dilation is applied.
- Plot a point on Circle \(d\). Label the point \(Q\). Plot a point that the dilation takes to \(Q\).
Problem 2
Here is triangle \(ABC\).
![Triangle A B C and point P on a circular grid. The coordinates of triangle A B C are A(negative 2 comma 2), B(negative 4 comma negative 1) and C(3 comma negative 1) and of point P at P(0 comma 0).](https://cms-im.s3.amazonaws.com/EuiRajwScmuwfumv9qAb3rMf?response-content-disposition=inline%3B%20filename%3D%228-8.2.A2.newPP.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.A2.newPP.Image.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001034Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=954b07d9446e8422c5825c89680b60e95411d4453c0b2c60828bb9e733bc3927)
- Dilate each vertex of triangle \(ABC\) using \(P\) as the center of dilation and a scale factor of 2. Draw the triangle connecting the three new points.
- Dilate each vertex of triangle \(ABC\) using \(P\) as the center of dilation and a scale factor of \(\frac 1 2\). Draw the triangle connecting the three new points.
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Measure the longest side of each of the three triangles. What do you notice?
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Measure the angles of each triangle. What do you notice?
Problem 3
Describe a rigid transformation that you could use to show the polygons are congruent.
![Triangles A B C and D E F.](https://cms-im.s3.amazonaws.com/xwXHVhqUtPxzYfCBDhhEymvc?response-content-disposition=inline%3B%20filename%3D%228-8.1.C.PP.Image.08.5b.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.C.PP.Image.08.5b.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001034Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b24c35d6d29c9f966b42e4ab7d4fcb3a2b138b9d7a6eb92e0cf691c9bfc2c522)
Problem 4
The line has been partitioned into three angles.
![A straight line with two rays coming out of a single point.](https://cms-im.s3.amazonaws.com/dc5os7EpTNp9H4DanFhYwXFt?response-content-disposition=inline%3B%20filename%3D%228-8.1.D.PP.Image.04.5.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.D.PP.Image.04.5.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001034Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c1f85bc42b7223db688ded409b7c520cc33f127f77cbc2ae82b02035a854312a)
Is there a triangle with these three angle measures? Explain.