Question

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Explanation:

Define dilation and its properties

A dilation is a transformation that changes the size of a figure but not its shape. It is an enlargement or reduction of a figure with respect to a fixed point, called the center of dilation. Because it changes the size, a dilation is an example of a non-rigid transformation (or similarity transformation) in that it does not preserve congruency. Instead, a dilation produces similar figures, where all corresponding angles are congruent and all corresponding sides are proportional.

Define scale factor and its conditions

The scale factor indicates how much the figure will enlarge or reduce. The variable commonly used for the scale factor is \(k\).

• When \(k > 1\), the dilation is an enlargement.
• When \(0 < k < 1\), the dilation is a reduction.

State the coordinate rule for dilation

If \(P(x, y)\) is the preimage of a point, then its image after a dilation centered at the origin \((0, 0)\) with scale factor \(k\) will follow the rule:

$$(x, y) ightarrow (kx, ky)$$

Solve Example 1

The problem asks to graph and label the figure and its image under a dilation with the origin as the center.

• Preimage: Triangle \(RST\) with vertices \(R(-5, 1)\), \(S(-3, 4)\), and \(T(2, -1)\).
• Scale factor: \(k = 2\).

Applying the rule \((x, y)
ightarrow (2x, 2y)\):

• \(R' = (2 \cdot (-5), 2 \cdot 1) = (-10, 2)\)
• \(S' = (2 \cdot (-3), 2 \cdot 4) = (-6, 8)\)
• \(T' = (2 \cdot 2, 2 \cdot (-1)) = (4, -2)\)

Solve Example 2

• Preimage: Rectangle \(ABCD\) with vertices \(A(-3, 0)\), \(B(1, 2)\), \(C(2, 0)\), and \(D(-2, -2)\).
• Scale factor: \(k = 3\).

Applying the rule \((x, y)
ightarrow (3x, 3y)\):

• \(A' = (3 \cdot (-3), 3 \cdot 0) = (-9, 0)\)
• \(B' = (3 \cdot 1, 3 \cdot 2) = (3, 6)\)
• \(C' = (3 \cdot 2, 3 \cdot 0) = (6, 0)\)
• \(D' = (3 \cdot (-2), 3 \cdot (-2)) = (-6, -6)\)

Answer:

Dilation Section

• A dilation is an enlargement or reduction of a figure with respect to a fixed point, called the center of dilation.
• A dilation is an example of a non-rigid transformation (or similarity transformation) in that it does not preserve congruency.
• A dilation produces similar figures.

Scale Factor Section

• Variable for scale factor: \(k\)
• When \(k > 1\), the dilation is an enlargement.
• When \(0 < k < 1\), the dilation is a reduction.

Dilation Rule Section

• Rule: \((x, y)

ightarrow\) \((kx, ky)\)

Examples Section

Question 1

• Preimage Vertices: \(R(-5, 1)\), \(S(-3, 4)\), \(T(2, -1)\)
• Dilation Rule (\(k = 2\)): \((x, y)

ightarrow (2x, 2y)\)

• Image Vertices:
• \(R'(-10, 2)\)
• \(S'(-6, 8)\)
• \(T'(4, -2)\)

Question 2

• Preimage Vertices: \(A(-3, 0)\), \(B(1, 2)\), \(C(2, 0)\), \(D(-2, -2)\)
• Dilation Rule (\(k = 3\)): \((x, y)

ightarrow (3x, 3y)\)

• Image Vertices:
• \(A'(-9, 0)\)
• \(B'(3, 6)\)
• \(C'(6, 0)\)
• \(D'(-6, -6)\)