Let’s solve each dilation problem step by step. Remember: a
dilation with scale factor *k* centered at the origin means you multiply
both x and y coordinates of every point by *k*. If *k* is between 0 and 1, the figure gets smaller. If *k* > 1, it gets bigger.
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Problem 1: Dilate △ABC with scale factor ½, center at origin.
Original points (from graph):
- A = (-2, 4)
- B = (-2, 1)
- C = (1, 1)
Multiply each coordinate by ½:
→ A’ = (-2 × ½, 4 × ½) =
(-1, 2)
→ B’ = (-2 × ½, 1 × ½) =
(-1, 0.5)
→ C’ = (1 × ½, 1 × ½) =
(0.5, 0.5)
Plot these new points and connect them to form the dilated triangle.
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Problem 2: Dilate △MNP with scale factor ⅓, center at origin.
Original points (from graph):
- M = (3, -2)
- N = (6, -2)
- P = (5, -4)
Multiply each coordinate by ⅓:
→ M’ = (3 × ⅓, -2 × ⅓) =
(1, -2/3) ≈ (1, -0.67)
→ N’ = (6 × ⅓, -2 × ⅓) =
(2, -2/3) ≈ (2, -0.67)
→ P’ = (5 × ⅓, -4 × ⅓) =
(5/3, -4/3) ≈ (1.67, -1.33)
Plot these and connect for the image.
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Problem 3: Dilate △JKL with scale factor ⅔, center at origin.
Original points (from graph):
- J = (-3, 3)
- K = (-3, 0)
- L = (0, 0)
Multiply each coordinate by ⅔:
→ J’ = (-3 × ⅔, 3 × ⅔) =
(-2, 2)
→ K’ = (-3 × ⅔, 0 × ⅔) =
(-2, 0)
→ L’ = (0 × ⅔, 0 × ⅔) =
(0, 0)
Plot and connect — notice L stays at origin since multiplying 0 by anything is still 0.
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Problem 4: Dilate △PQR with scale factor 3, center at origin.
Original points (from graph):
- P = (1, 1)
- Q = (2, 1)
- R = (1.5, 2) ← midpoint above base PQ
Multiply each coordinate by 3:
→ P’ = (1 × 3, 1 × 3) =
(3, 3)
→ Q’ = (2 × 3, 1 × 3) =
(6, 3)
→ R’ = (1.5 × 3, 2 × 3) =
(4.5, 6)
Plot these — this triangle will be much larger!
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✔ All calculations checked. Coordinates multiplied correctly by given scale factors.
Final Answer:
The images after dilation are formed by plotting the following transformed vertices:
1. △A’B’C’: A’(-1, 2), B’(-1, 0.5), C’(0.5, 0.5)
2. △M’N’P’: M’(1, -2/3), N’(2, -2/3), P’(5/3, -4/3)
3. △J’K’L’: J’(-2, 2), K’(-2, 0), L’(0, 0)
4. △P’Q’R’: P’(3, 3), Q’(6, 3), R’(4.5, 6)
Parent Tip: Review the logic above to help your child master the concept of dilations practice worksheet answers.
