Geometry Worksheet -- Dilations Worksheet - Free Printable

Let’s solve each dilation problem step by step. We’ll find the new coordinates of each triangle’s vertices after applying the given scale factor and center of dilation.

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Problem 1: Dilation scale = 1/2, center D(1,0)
Original points (from graph):
A(4,5), B(2,-2), C(4,-2)
Center D(1,0)

To dilate a point P(x,y) with center D(a,b) and scale k:
New x = a + k*(x - a)
New y = b + k*(y - b)

For A(4,5):
x’ = 1 + (1/2)*(4-1) = 1 + 1.5 = 2.5
y’ = 0 + (1/2)*(5-0) = 0 + 2.5 = 2.5 → A’(2.5, 2.5)

For B(2,-2):
x’ = 1 + (1/2)*(2-1) = 1 + 0.5 = 1.5
y’ = 0 + (1/2)*(-2-0) = 0 -1 = -1 → B’(1.5, -1)

For C(4,-2):
x’ = 1 + (1/2)*(4-1) = 1 + 1.5 = 2.5
y’ = 0 + (1/2)*(-2-0) = -1 → C’(2.5, -1)

✔ Check: All points moved halfway toward D(1,0). Makes sense for scale 1/2.

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Problem 2: Dilation scale = 1/4, center D(3,-4)
Original points:
A(-1,4), B(3,-8), C(-1,0)
Center D(3,-4)

For A(-1,4):
x’ = 3 + (1/4)*(-1-3) = 3 + (1/4)*(-4) = 3 -1 = 2
y’ = -4 + (1/4)*(4 - (-4)) = -4 + (1/4)(8) = -4 + 2 = -2 → A’(2, -2)

For B(3,-8):
x’ = 3 + (1/4)*(3-3) = 3 + 0 = 3
y’ = -4 + (1/4)*(-8 - (-4)) = -4 + (1/4)(-4) = -4 -1 = -5 → B’(3, -5)

For C(-1,0):
x’ = 3 + (1/4)*(-1-3) = 3 -1 = 2
y’ = -4 + (1/4)*(0 - (-4)) = -4 + 1 = -3 → C’(2, -3)

✔ Check: Points are much closer to D — correct for scale 1/4.

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Problem 3: Dilation scale = 2, center D(5,4)
Original points:
A(6,1), B(4,3), C(5,4) ← Wait! C is at (5,4)? That’s the center!

If a point is at the center, it doesn’t move. So C’ = C = (5,4)

For A(6,1):
x’ = 5 + 2*(6-5) = 5 + 2*1 = 7
y’ = 4 + 2*(1-4) = 4 + 2*(-3) = 4 -6 = -2 → A’(7, -2)

For B(4,3):
x’ = 5 + 2*(4-5) = 5 + 2*(-1) = 5 -2 = 3
y’ = 4 + 2*(3-4) = 4 + 2*(-1) = 4 -2 = 2 → B’(3, 2)

C’(5,4) — unchanged

✔ Check: Points doubled distance from center — looks right.

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Problem 4: Dilation scale = 4, center D(1,3)
Original points:
A(1,5), B(1,3), C(2,3)
Note: B is at center (1,3) → stays same.

For A(1,5):
x’ = 1 + 4*(1-1) = 1 + 0 = 1
y’ = 3 + 4*(5-3) = 3 + 8 = 11 → A’(1,11)

For C(2,3):
x’ = 1 + 4*(2-1) = 1 + 4 = 5
y’ = 3 + 4*(3-3) = 3 + 0 = 3 → C’(5,3)

B’(1,3) — unchanged

✔ Check: A went up 2 units → now 8 units up; C went right 1 → now 4 units right. Correct.

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Problem 5: Dilation scale = 1/2, center D(1,-5)
Original points:
A(-3,1), B(-3,-7), C(3,-3)
Center D(1,-5)

For A(-3,1):
x’ = 1 + (1/2)*(-3-1) = 1 + (1/2)(-4) = 1 -2 = -1
y’ = -5 + (1/2)*(1 - (-5)) = -5 + (1/2)(6) = -5 + 3 = -2 → A’(-1, -2)

For B(-3,-7):
x’ = 1 + (1/2)*(-3-1) = 1 -2 = -1
y’ = -5 + (1/2)*(-7 - (-5)) = -5 + (1/2)(-2) = -5 -1 = -6 → B’(-1, -6)

For C(3,-3):
x’ = 1 + (1/2)*(3-1) = 1 + 1 = 2
y’ = -5 + (1/2)*(-3 - (-5)) = -5 + (1/2)(2) = -5 +1 = -4 → C’(2, -4)

✔ Check: All points halfway to D — good.

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Problem 6: Dilation scale = 1/3, center D(-1,-3)
Original points:
A(-2,1), B(2,-5), C(-2,-3)
Note: C is at (-2,-3) — not center. Center is (-1,-3)

For A(-2,1):
x’ = -1 + (1/3)*(-2 - (-1)) = -1 + (1/3)(-1) = -1 - 1/3 = -4/3 ≈ -1.333
y’ = -3 + (1/3)*(1 - (-3)) = -3 + (1/3)(4) = -3 + 4/3 = -5/3 ≈ -1.667 → A’(-4/3, -5/3)

For B(2,-5):
x’ = -1 + (1/3)*(2 - (-1)) = -1 + (1/3)(3) = -1 +1 = 0
y’ = -3 + (1/3)*(-5 - (-3)) = -3 + (1/3)(-2) = -3 - 2/3 = -11/3 ≈ -3.667 → B’(0, -11/3)

For C(-2,-3):
x’ = -1 + (1/3)*(-2 - (-1)) = -1 + (1/3)(-1) = -4/3
y’ = -3 + (1/3)*(-3 - (-3)) = -3 + 0 = -3 → C’(-4/3, -3)

✔ Check: All points pulled 1/3 of the way toward D — correct.

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Final Answer:

1) A’(2.5, 2.5), B’(1.5, -1), C’(2.5, -1)
2) A’(2, -2), B’(3, -5), C’(2, -3)
3) A’(7, -2), B’(3, 2), C’(5, 4)
4) A’(1, 11), B’(1, 3), C’(5, 3)
5) A’(-1, -2), B’(-1, -6), C’(2, -4)
6) A’(-4/3, -5/3), B’(0, -11/3), C’(-4/3, -3)