Dilations from any center, and what a dilation does to lines

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   A dilation sends Kira's umbrella points A(4, 1) to A′(8, 5) and B(3, 2) to B′(5, 8). Which center and scale factor fit?

   • Center (2, -1), scale factor 1/3
   • Center (6, 3), scale factor 2
   • Center (4, 1), scale factor 3
   • Center (2, -1), scale factor 3
2. 2.

   Line l is any line, possibly through P and possibly not. A dilation about point P maps line ℓ to line ℓ′. Noah lists what is guaranteed. Which statement is correct?

   • ℓ′ is always parallel to ℓ and always a different line from ℓ
   • ℓ′ always passes through P
   • ℓ′ is always the same line as ℓ, since dilations only stretch lengths
   • ℓ′ is parallel to ℓ, and ℓ′ is always ℓ itself when ℓ passes through P
3. 3.

   Zee's leaf triangle is dilated by scale factor 5/2. How do the image's perimeter and area compare with the original's?

   • Both unchanged, since the shape is the same
   • Perimeter 25/4 times as large; area 5/2 times as large
   • Perimeter 5/2 times as large; area 25/4 times as large
   • Perimeter 5/2 times as large; area 5/2 times as large
4. 4.

   Bo's circle has center (2, -1) and radius 3. A dilation about the origin with scale factor 2 is applied. What is the image circle?

   • Center (4, -2), radius 5
   • Center (4, -2), radius 3
   • Center (4, -2), radius 6
   • Center (2, -1), radius 6
5. 5.

   Tala dilates segment AB with A(6, 3) and B(6, -5) by scale factor 1/2 about P(-2, 3), which is not on AB. Where are A′ and B′?

   • A′(4, 0) and B′(4, -4)
   • A′(2, 3) and B′(2, -1)
   • A′(3, 3/2) and B′(3, -5/2)
   • A′(13/2, 7/2) and B′(13/2, -9/2)
6. 6.

   Milo dilates by 2 about the origin, then by 1/2 about P(4, 0). Which single transformation does the pair equal?

   • The identity, since 2 × 1/2 = 1 and nothing moves
   • A translation 4 units right
   • A translation 2 units right
   • A translation 2 units left
7. 7.

   The center is not given. A dilation with scale factor 3 sends Pia's boat point A(2, 1) to A′(6, 3). Where does B(4, 5) go?

   • (12, 15)
   • (7, 8)
   • (8, 7)
   • (12, 5)
8. 8.

   Ren claims A is always the midpoint of PA′. With P(1, 1) and A(4, 3), Ren tests scale factor 3. What does the test show?

   • False: with factor 3, A′ = (7, 6) and the midpoint of PA′ is (4, 3.5), not A
   • True: it works when the factor is 2, so it works for any other factor too, such as 3
   • False: with factor 3, A′ = (10, 7) and the midpoint of PA′ is (5.5, 4), not A
   • True: A′ lies on line PA, so A is a midpoint
9. 9.

   A dilation with scale factor 3 about C(1, 0) sends line m to y = 2x + 7. Esme works back. What was m?

   • y = 2x + 21
   • y = 2x + 7/3
   • y = 2x + 4
   • y = 2x + 1
10. 10.

    Ari's projector lamp is the center. A 3 cm wide slide is 20 cm from the lamp, and the screen is 100 cm from it. How wide is the image?

    • 240 cm, using scale factor 80
    • 15 cm, using scale factor 5
    • 0.6 cm, using scale factor 1/5
    • 5 cm, the scale factor itself