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Enlargement Concept
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Today we will learn about enlargements in geometry. What do you think happens when we enlarge a shape?
The shape gets bigger, but it stays the same shape, right?
Exactly! That's a key point. The size changes, but the shape remains proportional. How do we describe enlargements mathematically?
We use a scale factor, don't we?
That's correct, Student_2! The scale factor tells us how much to multiply the dimensions of the original shape. Can anyone give an example of a scale factor?
If we have a scale factor of 3, then every length in the shape is multiplied by 3!
Well done! Remember this; it applies to each vertex. In our next session, we'll apply this knowledge to solve practice problems.
Practice Problem 1: Enlarging a Rectangle
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Letโs take rectangle EFGH with vertices E(1, 1), F(3, 1), G(3, 2), and H(1, 2). How do we enlarge it by a scale factor of 3?
We multiply each coordinate by the scale factor, right?
Exactly! So what would the new coordinates be?
For E, it would be (1 * 3, 1 * 3) = (3, 3). F would be (3 * 3, 1 * 3) = (9, 3), and so on!
Great job! What are the coordinates for G and H?
G would be (9, 6) and H would be (3, 6).
Awesome! So, the new vertices are E'(3, 3), F'(9, 3), G'(9, 6), and H'(3, 6). Remember that enlargements keep the shape but alter the size. Letโs practice more with another problem.
Practice Problem 2: Enlarge a Triangle
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Now we have triangle UVW with vertices U(-2, 4), V(-6, 4), and W(-2, 8). How would we enlarge this triangle by a scale factor of 0.5?
The new points will be half the distance from the origin!
Correct! What do you get when you apply the scale factor?
For U, it will be (-2 * 0.5, 4 * 0.5) = (-1, 2). V becomes (-3, 2), and W becomes (-1, 4).
Well done! Now you have correctly calculated the new vertices after the enlargement. Reducing a shape means getting closer to the center!
Practice Problem 3: Negative Scale Factor
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Let's say we enlarge triangle MNO with vertices M(1, 2), N(3, 2), O(2, 4) by a scale factor of -2. What happens here?
It will flip to the other side of the center while becoming larger!
Exactly! What do the new coordinates become?
For M, itโs (-1 * 2, -2 * 2) = (-2, -4). N would be (-6, -4) and O would be (-4, -8).
Correct! So, M'(-2, -4), N'(-6, -4), O'(-4, -8). When working with negative scale factors, itโs crucial to keep track of the new positioning. Great work today!
Introduction & Overview
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Quick Overview
Standard
In this section, students engage with practice problems that apply concepts of enlargements, scale factors, and coordinate transformations. By working through these exercises, they reinforce their understanding of how shapes can be resized while retaining their corresponding proportions and angles.
Detailed
Detailed Summary
In this section, we delve into practice problems associated with the concept of enlargements and dilations within geometric transformations. An enlargement is a transformation that modifies the size of a shape while preserving its overall shape. The key elements in these transformations include the center of enlargement and the scale factor, which determines how much larger or smaller the image becomes in comparison to the original shape. Students are tasked with various exercises that encourage them to apply the formulas for enlargements, calculate image coordinates for given vertices, and analyze the relationships between the dimensions of original and transformed figures. Through these practice problems, they will enhance their geometric problem-solving skills, gain confidence in applying transformation concepts, and understand the significance of scale and proportions in geometry.
Audio Book
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Problem 1: Enlarging a Rectangle
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- Enlarge rectangle EFGH with vertices E(1, 1), F(3, 1), G(3, 2), and H(1, 2) by a scale factor of 3, center at the origin. Write the image coordinates.
Detailed Explanation
To enlarge a rectangle using a scale factor, you need to multiply the coordinates of each vertex by that scale factor. Here, we're enlarging the rectangle EFGH with vertices at E(1, 1), F(3, 1), G(3, 2), and H(1, 2) by a scale factor of 3.
- Coordinates of E: E(1, 1) becomes (3 * 1, 3 * 1) = E'(3, 3).
- Coordinates of F: F(3, 1) becomes (3 * 3, 3 * 1) = F'(9, 3).
- Coordinates of G: G(3, 2) becomes (3 * 3, 3 * 2) = G'(9, 6).
- Coordinates of H: H(1, 2) becomes (3 * 1, 3 * 2) = H'(3, 6).
Thus, the coordinates of the enlarged rectangle E'F'G'H' are E'(3, 3), F'(9, 3), G'(9, 6), and H'(3, 6).
Examples & Analogies
Imagine you're taking a photo of a rectangle and you want to zoom in on it. Just like how you would stretch the image to cover more space on your screenโor make it largerโby adjusting the zoom settings, we applied a scale factor to the rectangle's coordinates to 'zoom' on the shape in the coordinate plane.
Problem 2: Enlarge a Triangle
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- Enlarge triangle UVW with vertices U(-2, 4), V(-6, 4), and W(-2, 8) by a scale factor of 1/2, center at the origin. Write the image coordinates.
Detailed Explanation
In this problem, we're reducing the size of triangle UVW with a scale factor of 1/2. This means we will multiply each vertex's coordinates by 1/2.
- Coordinates of U: U(-2, 4) becomes (-2 * 1/2, 4 * 1/2) = U'(-1, 2).
- Coordinates of V: V(-6, 4) becomes (-6 * 1/2, 4 * 1/2) = V'(-3, 2).
- Coordinates of W: W(-2, 8) becomes (-2 * 1/2, 8 * 1/2) = W'(-1, 4).
The new coordinates of the reduced triangle U'V'W' are U'(-1, 2), V'(-3, 2), and W'(-1, 4).
Examples & Analogies
Think of a drawing on a piece of paper. If you wanted a smaller version of the same drawing, you might place a transparency over the original and scale it down by making sure that all points on the original drawing are halved. This is similar to what we did by applying a scale factor of 1/2 to the triangle's coordinates.
Problem 3: Enlarging a Point with Negative Scale Factor
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- Point J(4, 0) is enlarged by a scale factor of -1.5 with the center at the origin. What are the coordinates of J'?
Detailed Explanation
To enlarge a point using a negative scale factor, we apply the scale factor to each coordinate while also considering the effect of the negatives. Here, we're enlarging the point J(4, 0) by -1.5:
- Coordinates of J: J(4, 0) becomes (4 * -1.5, 0 * -1.5) = J'(-6, 0).
This means that the point has not only moved twice its distance, but also on the opposite side of the origin. The coordinates of J' are (-6, 0).
Examples & Analogies
Imagine you're standing at a certain point on a line and looking straight ahead. Now, if you step backwards past your original spot further away, you'd be extending in the opposite direction. This is essentially how the negative scale factor worksโit flips you to the other side of the origin while also scaling your distance.
Problem 4: Areas Relating to Scale Factor
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- A square has vertices at (1,1), (3,1), (3,3), (1,3). If it is enlarged by a scale factor of 4 with the center at the origin, what is the area of the original square and the area of the image square? How do their areas relate to the scale factor? (Hint: Area scale factor is k squared).
Detailed Explanation
First, we need to calculate the area of the original square. The side length of the original square is the distance between any two adjacent vertices. In this case, the distance from (1, 1) to (1, 3) is 2 units (since the y-coordinates are the same and we just subtract the x-coordinates). Therefore, the area of the original square is 2 * 2 = 4 square units.
Now, after enlarging the square by a scale factor of 4, the new side length becomes 2 * 4 = 8 units. The area of the enlarged square is 8 * 8 = 64 square units.
To see how the areas relate to the scale factor, since the area scale factor is k squared, we calculate k squared: 4 * 4 = 16. This means the area of the image square is 16 times greater than the area of the original square! This verifies our understanding of how scale factors work in terms of area.
Examples & Analogies
Think about how when you enlarge or shrink a photo significantly, the physical area it occupies changes exponentially rather than linearly. For instance, a photo that starts out being 4 inches by 4 inches, when enlarged to 16 inches by 16 inches, doesnโt just get larger; it now fills an entire wall space, demonstrating how one action can create an enormous change!
Definitions & Key Concepts
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Key Concepts
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Enlargement: A geometric transformation that changes the size of a shape while keeping its overall form.
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Scale Factor: The ratio that expresses how much to multiply the dimensions of an object in enlargements.
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Center of Enlargement: The reference point from which all vertices' coordinates are scaled.
Examples & Real-Life Applications
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Examples
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Example 1: Enlarge triangle ABC with vertices A(1, 1), B(2, 2), C(3, 3) by a scale factor of 2, resulting in A'(2, 2), B'(4, 4), C'(6, 6).
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Example 2: Given rectangle JKL, vertices J(2, 2), K(4, 2), L(4, 3), and center at origin, enlarge it by a scale factor of 0.5, resulting in J'(1, 1), K'(2, 1), L'(2, 1.5).
Memory Aids
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๐ต Rhymes Time
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When enlarged, shapes go big or small, Scale factors measure, they change it all!
๐ Fascinating Stories
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Imagine a photographer zooming in on a tiny flower in a garden; the flower remains a flower, just much bigger - this is how enlargements work!
๐ง Other Memory Gems
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EES: Enlargement Equals Scale. Remember this to relate enlargement with its scale factor.
๐ฏ Super Acronyms
ECS
- Enlargement
- Coordinates
- Scale - key ideas in studying transformations.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Enlargement
Definition:
A transformation that changes the size of a geometric figure while preserving its shape.
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Term: Scale Factor
Definition:
A ratio that describes how much a figure is enlarged or reduced.
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Term: Coordinates
Definition:
A pair of numbers used to determine the position of points on the coordinate plane.
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Term: Center of Enlargement
Definition:
A fixed point from which a shape is enlarged or reduced.