Geometric Transformations: Dilations

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Concept Review

Dilations: The Shape-Stretching Magic

Imagine you're using a photocopier to enlarge a photograph. The picture gets bigger, but something stays exactly the same—the angles in every corner remain unchanged. This is the magic of dilations.

A dilation is like having a mathematical zoom lens. It stretches or shrinks a shape by multiplying all distances from a center point by the same number (called the scale factor). But here's what makes dilations special: while side lengths change dramatically, every single angle measurement stays perfectly preserved.

Seeing Dilation in Action

Let's watch a triangle transform. Start with triangle ABC where:

•Side AB = 4 units, angle A = 60°
•Side BC = 6 units, angle B = 90°
•Side AC = 8 units, angle C = 30°

Now apply a dilation with scale factor 2.5 from the origin. Every distance from the center gets multiplied by 2.5, creating triangle A'B'C' where:

•Side A'B' = 10 units, angle A' = still 60°
•Side B'C' = 15 units, angle B' = still 90°
•Side A'C' = 20 units, angle C' = still 30°

🔑 Mind-Bending Truth

You can stretch a shape to be 10 times bigger or shrink it to half size, but the angles refuse to budge even a single degree. It's like the angles have a built-in resistance to change—they're the unchangeable DNA of the shape's identity.

Scale Factors: The Dilation Controller

The scale factor is your remote control for dilations:

k > 1

Enlargement

Shape grows bigger

0 < k < 1

Reduction

Shape shrinks smaller

k = 1

Identity

No change at all

Key Takeaway: Just like that photocopier preserves the angles in your photograph while changing its size, dilations are the mathematical tool that changes everything about a shape's dimensions except the one thing that defines its essential character—its angles. In the world of transformations, dilations are the ultimate shape-preserving size-changers.

Sample questions

1. When a triangle is dilated by a scale factor of 2, what happens to its side lengths?

○ They stay the same

○ They are divided by 2

○ They are squared

✓ They are multiplied by 2

Answer: They are multiplied by 2 — Dilations multiply all side lengths by the scale factor.

2. After a dilation with scale factor 0.5, what happens to the angle measures of a pentagon?

✓ They stay the same

○ They are multiplied by 0.5

○ They are divided by 0.5

○ They become half of their original measure

Answer: They stay the same — Dilations preserve angle measures; only side lengths change.

3. A square is dilated by a scale factor of 3. Which statement is true?

○ The image is a rectangle

✓ The image is a square with sides 3 times longer

○ The image is a rhombus

○ The image has the same area as the original

Answer: The image is a square with sides 3 times longer — Dilations preserve shape (angles) but change size. A square remains a square.

Skills in this topic

Understand that dilations alter side lengths but preserve angle measures
Dilate a polygon on the coordinate plane from the origin given a scale factor greater than 1 (enlargement)
Dilate a polygon on the coordinate plane from the origin given a scale factor between 0 and 1 (reduction)
Identify the algebraic rule for a dilation (x,y) → (kx, ky)
Calculate the scale factor of a dilation given the pre-image and image

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