Geometry - Unit 3 Day 3

Unit 3: Symmetry & Transformations

Day 3: Translations & Dilations

Essential Question: How do we slide and resize figures on the coordinate plane?

Today's Goals

🎯 I will be able to translate a figure given a coordinate rule.

🎯 I will be able to dilate a figure from the origin given a scale factor.

Key Vocabulary

  • Translation: A transformation that slides a figure to a new position without turning it.
  • Dilation: A transformation that resizes a figure, making it larger or smaller.
  • Scale Factor (k): The number you multiply the coordinates by during a dilation.
  • Enlargement: A dilation where the figure gets bigger (k > 1).
  • Reduction: A dilation where the figure gets smaller (0 < k < 1).

I Do: Translating a Figure

A translation slides every point the same distance and direction. Let's translate this triangle right 6 units and down 2 units.

A(-8,-2) A'(-2,-4)

Preimage A: (-8, -2)

Image A': (-2, -4)

We added 6 to the x-coordinate (-8 + 6 = -2) and subtracted 2 from the y-coordinate (-2 - 2 = -4).

Rule: (x, y) → (x + 6, y - 2)

We Do: Applying a Translation Rule

Let's translate the trapezoid using the rule (x, y) → (x - 5, y + 7). What does this rule tell us to do?

A(2, -8) → A'(-3, -1)

B(8, -8) → B'(3, -1)

C(6, -4) → C'(1, 3)

D(4, -4) → D'(-1, 3)

The rule means slide left 5 and up 7.

You Do: Dilation by k = 2

Now it's your turn! A dilation resizes a figure. Multiply every coordinate by the scale factor, k. Here, k=2, so it's an enlargement.

A(2, 1) → A':(, )

B(5, 1) → B':(, )

C(3, 4) → C':(, )

Rule: (x, y) → (2x, 2y)

I Do: Dilation by k = \(\frac{1}{2}\)

When the scale factor is between 0 and 1, the image is a reduction (it gets smaller). Let's dilate by k = 0.5.

Preimage: (-8, -2)

Image: (-4, -1)

We multiply each coordinate by 0.5.

Rule: (x, y) → (0.5x, 0.5y)

We Do: Find the Scale Factor

Here, the dilation made the figure smaller. We call this a reduction. Let's find the scale factor (k) using the same rule as before.

Preimage A(-12, 4) Image A'(-3, 1)

The Rule:

k = \(\frac{\text{New}}{\text{Old}}\)

Step-by-Step:

  1. Notice the image is smaller, so we expect a scale factor between 0 and 1.
  2. Pick a matching pair of points:
    A(-12, 4) and A'(-3, 1).
  3. Compare their x-coordinates:
    k = \(\frac{-3}{-12}\) = \(\frac{1}{4}\) or 0.25
  4. Check with the y-coordinates:
    k = \(\frac{1}{4}\) = 0.25

The scale factor is k = \(\frac{1}{4}\). Since 0 < k < 1, this is a reduction.

You Do: Find the Scale Factor

Now it's your turn. This time the image is an enlargement. Use the same method to find the scale factor.

P(2, 1) P'(6, 3)

The Rule:

k = \(\frac{\text{New}}{\text{Old}}\)

Your Steps:

  1. Pick a matching pair of points:
    P(2, 1) and P'(, ).
  2. Compare the x-coordinates:
    k = \(\frac{\text{New X}}{\text{Old X}}\) = \(\frac{?}{?}\) =
  3. Check with the y-coordinates:
    k = \(\frac{\text{New Y}}{\text{Old Y}}\) = \(\frac{?}{?}\) =

The scale factor is k = . Since k > 1, this is an enlargement.

Transformation Rules Summary

Let's summarize today's rules. Remember, translations, reflections, and rotations are rigid motions (size and shape don't change). Dilation is not a rigid motion.

Translations

(x, y) → (x + a, y + b)

Slides right/left by 'a'

Slides up/down by 'b'

Dilations (from origin)

(x, y) → (kx, ky)

k > 1: Enlargement

0 < k < 1: Reduction

Important Note:

When a figure is dilated, the new figure is similar to the original, but not congruent (unless k=1).

Independent Practice

Level: Green

1. Point A(3, 8) is translated by the rule (x, y) → (x - 4, y + 2). What are the coordinates of A'?

2. Point B(10, -6) is dilated by a scale factor of k = 0.5. What are the coordinates of B'?

Level: Yellow

3. Triangle CDE has vertices C(0,0), D(2,4), and E(4,0). Find the coordinates of the image after a dilation with k = 3.

4. A point P(-3, 5) is translated to P'(2, 1). Write the coordinate rule for this translation.

Level: Red

5. A rectangle's image after a dilation has vertices at (0,0), (10,0), (10,6), and (0,6). If the scale factor was 2, what were the coordinates of the original preimage?

6. Is it possible for a translated figure to be in the same location as the original? If so, how? Explain.

✅ Exit Ticket

Answer the following questions to show what you've learned about translations and dilations.

1. A square is translated using the rule (x, y) → (x, y - 5). Describe this translation in words.

2. Dilate the point F(-9, 3) using a scale factor of k = \(\frac{1}{3}\). What are the coordinates of F'?

3. A triangle is dilated by a scale factor of 4. Is the new triangle congruent to the original? Why or why not?