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Understanding Enlargement and Scale Factor
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Welcome, everyone! Today, we will explore how we can change the size of shapes using a process called enlargement. Who can tell me what a scale factor is?
Isn't it the number you multiply by to resize a shape?
Exactly! A scale factor can be greater than 1 to enlarge a shape or between 0 and 1 to reduce it. What happens to the shape when we use a scale factor less than 1?
The shape becomes smaller, right?
Correct! If we take a triangle and apply a scale factor of 1/2, the new triangle will be half the size of the original. Letโs look at our example triangle ABC with coordinates A(4, 6), B(8, 2), and C(2, 2). If we use 1/2 as our scale factor, how do we find the new coordinates?
We multiply each coordinate by 1/2!
Thatโs right! This process helps us find the new vertices based on the scale factor. Let's calculate those together!
Applying the Scale Factor to Calculate New Coordinates
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Now that we understand the scale factor, let's calculate the new coordinates for triangle ABC when we apply a scale factor of 1/2. What will the coordinates for point A (4, 6) become?
That would be A'(2, 3) after multiplying by 1/2.
Great! How about point B (8, 2)?
B' would be (8 * 1/2, 2 * 1/2), so B' becomes (4, 1).
Exactly! And for C (2, 2)?
C' would be (1, 1)!
Perfect! So the new coordinates after applying the scale factor of 1/2 are A'(2, 3), B'(4, 1), and C'(1, 1). What do you notice about the shape of the new triangle compared to the original?
Itโs smaller, but it still looks the same, just proportionally reduced!
Understanding Similarity Through Enlargement
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Now that we've found our new triangleโs coordinates, letโs discuss how enlargement relates to similarity. Why do we say the new triangle is similar to the original triangle?
Because they have the same shape?
That's correct! Both triangles maintain the same angles. So, by applying a scale factor, we change the size while keeping the shape intact. Can anyone explain what โcorresponding anglesโ means?
It means that the angles of the two triangles match in size.
Right! In similar shapes, corresponding angles are equal, and the sides are proportional. This concept is essential when analyzing geometric figures. We use the idea of similarity a lot in real life, like with maps or models!
Introduction & Overview
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Quick Overview
Standard
In this section, we delve into enlargements using a fractional scale factor to transform triangles. Through examples, we demonstrate how vertices change position based on the specified scale factor, and we explore the implications of size reduction on the properties of the figure.
Detailed
Detailed Summary
This section focuses on the concept of enlargement (dilation) in geometric transformations, specifically applying a fractional scale factor to a triangle. The primary objective is to illustrate how a shape can be reduced in size while retaining its overall structure and proportions.
In geometric terms, a dilation is a transformation that alters the size of a figure based on a scale factor (k), which is a multiplier that proportions the distances from a specific point known as the center of enlargement.
- When the scale factor is greater than 1, the image enlarges.
- When the scale factor is between 0 and 1 (as in the examples here), the image reduces in size, leading to a shape that is smaller but retains the same overall proportions.
- The properties of similarity are preserved such that all corresponding angles remain equal while the sides of the resulting triangle are shorter in direct proportion to the scale factor.
Through a practical example involving triangle ABC, with vertices A(4, 6), B(8, 2), and C(2, 2), which is reduced using a scale factor of 1/2 centered at the origin (0, 0), the section illustrates step-by-step calculations to derive the new coordinates of the reduced triangle.
This provides not only a theoretical overview but also a practical understanding of the transformation's application in geometry. Additionally, it connects back to the broader theme of transformations and similarity, reinforcing analytical skills necessary for geometry.
Audio Book
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Enlargement Definition
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Enlarge triangle ABC with vertices A(4, 6), B(8, 2), and C(2, 2) by a scale factor of 1/2, center at the origin (0, 0).
Detailed Explanation
The problem states we need to enlarge (or reduce) triangle ABC using a scale factor of 1/2. A scale factor below 1 indicates a reduction in size. This means each vertex of the triangle will move closer to the origin, effectively halving the distance from the center.
Examples & Analogies
Think of a large pizza being sliced into smaller pieces. If you take one slice and reduce its size by half, it will become smaller, but maintain its shape as a slice. Similarly, we are reducing the triangle's size while keeping its overall shape intact.
Applying the Scale Factor
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โ Step 1: Apply the rule (0.5x, 0.5y) to each vertex.
โ A'(0.54, 0.56) = A'(2, 3)
โ B'(0.58, 0.52) = B'(4, 1)
โ C'(0.52, 0.52) = C'(1, 1)
Detailed Explanation
To apply the scale factor of 1/2 to each vertex of triangle ABC, we multiply the x and y coordinates of each vertex by 0.5. For example, A was at (4, 6) and after multiplying each coordinate by 0.5, it becomes A' at (2, 3). Similarly, we do this for B and C to find their new locations.
Examples & Analogies
Imagine you have a model of a building. When you want a smaller version of that building, you simply scale down every measurement (height, width). You take each dimension, like those from the original triangle, and reduce them by half to create a precise smaller version.
Plotting the New Triangle
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โ Step 2: Plot the image. The image triangle will be half the size of the object triangle.
Detailed Explanation
Once we have the new coordinates for each vertex after applying the scale factor, we plot these points on the coordinate plane. Doing this will visually display how the new triangle (A'B'C') has been reduced in size compared to the original triangle (ABC), clearly showing the transformation.
Examples & Analogies
Think of drawing a smaller version of a picture. You take the original size and carefully mark where each point would land if you were to make it half as big. Just like accurate measurements in drawing, these plotted points will help you visualize the transformation clearly.
Definitions & Key Concepts
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Key Concepts
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Scale Factor: Refers to the number that multiplies the dimensions of a shape during enlargement or reduction.
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Dilation: A transformation that changes the size of a shape while keeping its proportions.
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Center of Enlargement: The fixed point around which transformation takes place.
Examples & Real-Life Applications
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Examples
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Example of reducing triangle ABC with coordinates A(4, 6), B(8, 2), C(2, 2) using a scale factor of 1/2 results in new points A'(2, 3), B'(4, 1), C'(1, 1).
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If Triangle DEF is enlarged by a scale factor of 3, the new vertices would be three times the distance from the center.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
๐ต Rhymes Time
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Scaling down with one-half, your triangle shrinks in size, yet its angles donโt disguise!
๐ Fascinating Stories
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Imagine a small painter, who loves to shrink giant images of trees to fit into a tiny book, showing how shapes can get smaller but still tell the same story!
๐ง Other Memory Gems
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Remember, SAME for shapes: Similar shapes have Angles Same, Measure proportionately.
๐ฏ Super Acronyms
DILATION
- Dilate - Increase/Decrease size; Keep corresponding Angles and Proportions.
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 shape while keeping its shape the same; can be either an increase or a decrease in size.
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Term: Fractional Scale Factor
Definition:
A scale factor that is less than 1 which reduces the size of the shape.
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Term: Dilation
Definition:
A transformation that alters a shape by enlarging or reducing it based on a scale factor.
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Term: Center of Enlargement
Definition:
The fixed point in a plane around which the enlargement occurs.
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Term: Similar Figures
Definition:
Figures that have the same shape but may be different sizes; they maintain proportional sides and equal corresponding angles.