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

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   Luz's dilation has center P(1, -2). It sends D(4, 2) to D′(13, 14). What is the scale factor?

   • 1/4
   • 4
   • 3
   • 20
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 parallel to ℓ, and ℓ′ is always ℓ itself when ℓ passes through P
   • ℓ′ always passes through P
   • ℓ′ is always parallel to ℓ and always a different line from ℓ
   • ℓ′ is always the same line as ℓ, since dilations only stretch lengths
3. 3.

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

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

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

   • A translation 2 units left
   • A translation 2 units right
   • The identity, since 2 × 1/2 = 1 and nothing moves
   • A translation 4 units right
6. 6.

   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?

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

   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?

   • 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
   • False: with factor 3, A′ = (7, 6) and the midpoint of PA′ is (4, 3.5), not A
8. 8.

   Perimeter 14 cm, area 8 cm². Scale factor 1/3. A dilation with scale factor 1/3 makes Jun's star triangle have perimeter 14 cm and area 8 cm². What were the original perimeter and area?

   • Perimeter 14/3 cm, area 8/9 cm²
   • Perimeter 42 cm, area 24 cm²
   • Perimeter 42 cm, area 72 cm²
   • Perimeter 17 cm, area 11 cm²
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 + 1
   • y = 2x + 7/3
   • y = 2x + 4
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