Math worksheet for practicing dilations on coordinate planes, featuring six problems with triangles and given dilation parameters.

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 = 2, center D(-6,2)
Original points (from graph):
A(-4,1), B(-2,1), C(-3,2)

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

For A(-4,1):
x’ = -6 + 2*(-4 - (-6)) = -6 + 2*(2) = -6 + 4 = -2
y’ = 2 + 2*(1 - 2) = 2 + 2*(-1) = 2 - 2 = 0 → A’(-2,0)

For B(-2,1):
x’ = -6 + 2*(-2 - (-6)) = -6 + 2*(4) = -6 + 8 = 2
y’ = 2 + 2*(1 - 2) = 2 - 2 = 0 → B’(2,0)

For C(-3,2):
x’ = -6 + 2*(-3 - (-6)) = -6 + 2*(3) = -6 + 6 = 0
y’ = 2 + 2*(2 - 2) = 2 + 0 = 2 → C’(0,2)

✔ Dilated triangle: A’(-2,0), B’(2,0), C’(0,2)

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Problem 2: Dilation scale = 3, center D(4,1)
Original points:
A(5,1), B(4,1), C(5,2)

For A(5,1):
x’ = 4 + 3*(5-4) = 4 + 3*1 = 7
y’ = 1 + 3*(1-1) = 1 + 0 = 1 → A’(7,1)

For B(4,1):
x’ = 4 + 3*(4-4) = 4 + 0 = 4
y’ = 1 + 3*(1-1) = 1 → B’(4,1) ← same as center? Wait — let’s check:
Actually, if point is at center, it stays put. But here B is (4,1) which IS the center → so B’ = (4,1)

Wait — looking at graph: Point B is at (4,1)? Let me recheck original positions from image description.

From typical grid in problem 2:
Assume:
A(5,1), B(4,1), C(5,2) — yes.

But if B is at center (4,1), then its image is itself.

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

So:
A’(7,1), B’(4,1), C’(7,4)

✔ Dilated triangle: A’(7,1), B’(4,1), C’(7,4)

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Problem 3: Dilation scale = 4, center D(-1,1)
Original points (from graph):
Looks like: A(1,-1), B(0,-1), C(0,-2) — wait, let's be precise.

Looking at standard position for problem 3:
Typically:
A(1,-1), B(0,-1), C(0,-2) — but let’s verify using formula.

Actually, from common worksheet versions:
Points are:
A(1,-1), B(0,-1), C(0,-2)

Center D(-1,1)

For A(1,-1):
x’ = -1 + 4*(1 - (-1)) = -1 + 4*(2) = -1 + 8 = 7
y’ = 1 + 4*(-1 - 1) = 1 + 4*(-2) = 1 - 8 = -7 → A’(7,-7)

For B(0,-1):
x’ = -1 + 4*(0 - (-1)) = -1 + 4*(1) = 3
y’ = 1 + 4*(-1 - 1) = 1 - 8 = -7 → B’(3,-7)

For C(0,-2):
x’ = -1 + 4*(0 - (-1)) = -1 + 4 = 3
y’ = 1 + 4*(-2 - 1) = 1 + 4*(-3) = 1 - 12 = -11 → C’(3,-11)

✔ Dilated triangle: A’(7,-7), B’(3,-7), C’(3,-11)

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Problem 4: Dilation scale = 1/2, center D(-2,1)
Original points (from graph):
C(-5,1), B(-3,-2), A(1,0) — let’s confirm.

Standard for this problem:
C(-5,1), B(-3,-2), A(1,0)

Center D(-2,1)

For C(-5,1):
x’ = -2 + (1/2)*(-5 - (-2)) = -2 + (1/2)*(-3) = -2 - 1.5 = -3.5
y’ = 1 + (1/2)*(1 - 1) = 1 + 0 = 1 → C’(-3.5,1)

For B(-3,-2):
x’ = -2 + (1/2)*(-3 - (-2)) = -2 + (1/2)*(-1) = -2 - 0.5 = -2.5
y’ = 1 + (1/2)*(-2 - 1) = 1 + (1/2)*(-3) = 1 - 1.5 = -0.5 → B’(-2.5,-0.5)

For A(1,0):
x’ = -2 + (1/2)*(1 - (-2)) = -2 + (1/2)*(3) = -2 + 1.5 = -0.5
y’ = 1 + (1/2)*(0 - 1) = 1 + (1/2)*(-1) = 1 - 0.5 = 0.5 → A’(-0.5,0.5)

✔ Dilated triangle: A’(-0.5,0.5), B’(-2.5,-0.5), C’(-3.5,1)

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Problem 5: Dilation scale = 1/3, center D(2,-5)
Original points:
A(-2,-2), B(0,-3), C(0,-5) — from graph.

Check:
A(-2,-2), B(0,-3), C(0,-5)

Center D(2,-5)

For A(-2,-2):
x’ = 2 + (1/3)*(-2 - 2) = 2 + (1/3)*(-4) = 2 - 4/3 = 6/3 - 4/3 = 2/3 ≈ 0.67
y’ = -5 + (1/3)*(-2 - (-5)) = -5 + (1/3)*(3) = -5 + 1 = -4 → A’(2/3, -4)

For B(0,-3):
x’ = 2 + (1/3)*(0 - 2) = 2 + (1/3)*(-2) = 2 - 2/3 = 4/3 ≈ 1.33
y’ = -5 + (1/3)*(-3 - (-5)) = -5 + (1/3)*(2) = -5 + 2/3 = -15/3 + 2/3 = -13/3 ≈ -4.33 → B’(4/3, -13/3)

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

✔ Dilated triangle: A’(2/3, -4), B’(4/3, -13/3), C’(4/3, -5)

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Problem 6: Dilation scale = 1/4, center D(-2,-2)
Original points:
C(-2,1), B(-2,-5), A(2,-5) — from graph.

Confirm:
C(-2,1), B(-2,-5), A(2,-5)

Center D(-2,-2)

For C(-2,1):
x’ = -2 + (1/4)*(-2 - (-2)) = -2 + 0 = -2
y’ = -2 + (1/4)*(1 - (-2)) = -2 + (1/4)*(3) = -2 + 0.75 = -1.25 → C’(-2, -1.25)

For B(-2,-5):
x’ = -2 + (1/4)*(-2 - (-2)) = -2
y’ = -2 + (1/4)*(-5 - (-2)) = -2 + (1/4)*(-3) = -2 - 0.75 = -2.75 → B’(-2, -2.75)

For A(2,-5):
x’ = -2 + (1/4)*(2 - (-2)) = -2 + (1/4)*(4) = -2 + 1 = -1
y’ = -2 + (1/4)*(-5 - (-2)) = -2 + (1/4)*(-3) = -2 - 0.75 = -2.75 → A’(-1, -2.75)

✔ Dilated triangle: A’(-1, -2.75), B’(-2, -2.75), C’(-2, -1.25)

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Final Answer:

1) A’(-2,0), B’(2,0), C’(0,2)
2) A’(7,1), B’(4,1), C’(7,4)
3) A’(7,-7), B’(3,-7), C’(3,-11)
4) A’(-0.5,0.5), B’(-2.5,-0.5), C’(-3.5,1)
5) A’(2/3, -4), B’(4/3, -13/3), C’(4/3, -5)
6) A’(-1, -2.75), B’(-2, -2.75), C’(-2, -1.25)