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Dilations on the Coordinate Plane. A dilation is a transformation that changes the *size* of a figure but not its *shape*. It’s defined by:
- A
center of dilation (here, always the origin: `(0, 0)`)
- A
scale factor (`k`)
How to Dilate a Point:
To dilate any point `(x, y)` with scale factor `k` centered at the origin, multiply both coordinates by `k`:
>
New point = `(k·x, k·y)`
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## Problem 1: Graph ΔABC after dilation with scale factor 2 (already done for you — good!)
Original points (from graph):
- A(1, 1)
- B(3, 1)
- C(1, 4)
Dilated by scale factor 2:
- A’ = (2×1, 2×1) =
(2, 2)
- B’ = (2×3, 2×1) =
(6, 2)
- C’ = (2×1, 2×4) =
(2, 8)
✔ Matches the red triangle shown.
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## Problem 2: Graph ΔMNP after dilation with scale factor ½, centered at origin.
First, read original coordinates from the graph:
- M(-2, -1)
- N(0, 3)
- P(2, -1)
Apply scale factor
½ → Multiply each coordinate by 0.5:
- M’ = (½ × -2, ½ × -1) =
(-1, -0.5)
- N’ = (½ × 0, ½ × 3) =
(0, 1.5)
- P’ = (½ × 2, ½ × -1) =
(1, -0.5)
📌
Plot these points and connect them to form the dilated triangle.
This will be a smaller triangle inside the original, halfway between the original and the origin.
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## Problem 3: Graph ΔJKL after dilation with scale factor ¼, centered at origin.
Original points (from graph):
- J(-4, 0)
- K(-4, 6)
- L(0, 0)
Apply scale factor
¼ → Multiply each coordinate by 0.25:
- J’ = (¼ × -4, ¼ × 0) =
(-1, 0)
- K’ = (¼ × -4, ¼ × 6) =
(-1, 1.5)
- L’ = (¼ × 0, ¼ × 0) =
(0, 0) ← stays at origin!
📌
Plot these points and connect them.
This triangle will be much smaller, very close to the origin.
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## Problem 4: Graph ΔRST after dilation with scale factor 3, centered at origin.
Original points (from graph):
- R(-1, 0)
- S(0, 2)
- T(1, 0)
Apply scale factor
3 → Multiply each coordinate by 3:
- R’ = (3 × -1, 3 × 0) =
(-3, 0)
- S’ = (3 × 0, 3 × 2) =
(0, 6)
- T’ = (3 × 1, 3 × 0) =
(3, 0)
📌
Plot these points and connect them.
This triangle will be larger, stretched out from the origin.
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##
✔ Final Answer Summary:
Here are the new coordinates for each dilated triangle:
2. ΔMNP (scale factor ½):
- M’(-1, -0.5)
- N’(0, 1.5)
- P’(1, -0.5)
3. ΔJKL (scale factor ¼):
- J’(-1, 0)
- K’(-1, 1.5)
- L’(0, 0)
4. ΔRST (scale factor 3):
- R’(-3, 0)
- S’(0, 6)
- T’(3, 0)
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## 💡 Tip for Graphing:
- Use grid lines to accurately plot fractional or decimal coordinates.
- Remember: if scale factor > 1 → enlargement; if 0 < scale factor < 1 → reduction.
- All dilations here are centered at origin, so every point moves along the line connecting it to (0,0).
Let me know if you’d like help plotting these on graph paper or checking your drawing!
Parent Tip: Review the logic above to help your child master the concept of dilations worksheet.
