1 . Point Q is the image of point P under a dilation with center and scale factor 5. If PQ = 18 then what is OP?

2 . A dilation with scale factor -6 maps X to X' and Y to Y'. Given X'Y' = 20, find XY.

3a. We cut a regular heptagon ABCDEFG out of a piece of cardboard. How many different ways can we place the heptagon back in the cardboard, if we are allowed to rotate but not reflect the heptagon?

3b. How many different ways can we place the heptagon back in the cardboard, if we are allowed to rotate and reflect the heptagon?

4. A reflection maps A to B and C to C. If AB = 40, then what is the shortest possible length of AC?

5. In triangle ABC, AB = 5, BC = 8, and CA = 4

Triangle ABC is first rotated clockwise about A for 90 degrees to triangle AB_{1}C_{1}.

Triangle AB_{1}C_{1} is then reflected over to triangle A_{1}B_{1}C_{1}.

Triangle A_{1}B_{1}C_{1} is then translated 3 units to the right to triangle A_{2}B_{2}C_{2}.

Triangle A_{2}B_{2}C_{2} is then dilated about A to triangle A_{3}B_{3}C_{3} with scale factor -2.

What is the perimeter of triangle A_{3}B_{3}C_{3?}

_{6. }In the diagram below, Y is reflected over XZ to Y'. Find YY'.

7. In triangle ABC, D, E, and F are the midpoints of BC, CA, and AB, respectively. A dilation maps A to D, B to E, and C to F. What is the scale factor of this dilation?

katt09 May 2, 2023

#2**0 **

I will try to figure some of these out.

1. O is not given, neither is the center of the dilation.

2. A dilation divides all factors involved, including segment length, by its scale factor. So the answer is 20/-6, which you can simplify. A hint, though, whether a dilation is positive or negative only determines which side of the center of the dilation the result is on.

3a. 7 ways, since we can decide which point goes in any certain spot.

3b. 14 ways, since we can now reflect.

4. We can see that if A is 40 units away from B than it is 20 units away from the line. Given that since C goes to C it is on the line, how close can A be to C?

5. Most of the information does not matter since it does not change shape, only location. Use the dilation information I gave in problem two (just with scale factor -2 instead) to figure it out.

6. I don't know how to do this one.

7. If triangle ABC is equilateral, then the scale factor is -2. Otherwise, I don't think any dilation can do this.

gb1falcon May 3, 2023