Practice Problems 4.2
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Understanding Similarity
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Today, we're exploring the concept of similarity. Can anyone remind me what makes two figures similar?
Is it when they have the same shape but not necessarily the same size?
Exactly! They have the same shape, which means all corresponding angles are equal. Great job! Now, how about the sides?
The sides are in proportion, right?
Absolutely! The ratio of corresponding sidesβcalled the scale factorβremains constant. Remember, the key to identifying similarity is checking both angles and side lengths.
So, if I have a triangle and a smaller triangle that looks like it, I can measure the sides and angles to know if they're similar?
Correct! Always check the angles first since they confirm similarity. Let's move to some practice problems now.
Finding Scale Factor
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Letβs consider triangles LMN and PQR. If we know LM is 8 cm and PQ is 4 cm, what would the scale factor be?
The scale factor from LMN to PQR would be 8 divided by 4, so k = 2.
Correct! So, triangle PQR is half the size of triangle LMN. How would you find another side, like QR, given that MN is 10 cm?
I would use the scale factor! Since the scale is 2, QR would be 10 divided by 2, which is 5 cm.
Exactly! You can always determine unknown lengths by applying the scale factor. Let's try another example.
Proportions in Similar Shapes
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Now we will set up proportions to find unknown lengths. For trapezoid ABCD similar to trapezoid EFGH, if AB is 4, BC is 6, and EF is 10, how do we find FG?
We can set up the proportion, 4 over 10 equals 6 over FG.
Great! Now, can someone show me how to solve for FG using cross-multiplication?
Sure! If we cross-multiply, we get 4 * FG = 10 * 6, which is 4FG = 60. So FG would be 60 divided by 4, making FG 15.
Perfect! Setting up proportions is incredibly useful for dealing with similar shapes. Can anyone remember when we might use this in real life?
In architecture or design, we can create scaled models of buildings!
Exactly right! Understanding similarity and proportions has real-world applications.
Introduction & Overview
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Quick Overview
Standard
The section includes a variety of practice problems that require the application of concepts such as similarity, scale factors, and proportions to determine unknown lengths. It challenges students to use their understanding of geometric transformations in practical scenarios.
Detailed
In section 4.2, students engage with practice problems focusing on the application of similarity in geometry. They are tasked with determining the scale factor between similar triangles, applying knowledge of proportions to find unknown side lengths in various geometric figures. By solving these problems, learners gain a deeper understanding of the importance of similarity in geometric transformations, highlighting how shapes can be scaled while preserving their properties. This section is pivotal in connecting theoretical learning with practical application, reinforcing the concepts learned in previous chapters.
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Problem 1: Similar Triangles
Chapter 1 of 2
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Chapter Content
Triangle LMN is similar to triangle PQR. LM = 8 cm, MN = 10 cm, LN = 12 cm. PQ = 4 cm.
- Find the scale factor from triangle LMN to triangle PQR.
- Find the lengths of QR and PR.
Detailed Explanation
In this problem, you're given two similar triangles, LMN and PQR, and you need to determine their scale factor as well as the unknown side lengths.
1. Finding the Scale Factor (k): The scale factor can be found by comparing the lengths of the sides of the triangles. Since LM (8 cm) corresponds to PQ (4 cm), we calculate the scale factor:
k = (Length of a side in triangle PQR) / (Corresponding side in triangle LMN) = PQ / LM = 4 cm / 8 cm = 0.5.
This means that triangle PQR is half the size of triangle LMN.
- Finding QR: Since MN (10 cm) in LMN corresponds to QR in PQR, we can find QR by multiplying MN by the scale factor:
QR = MN * k = 10 cm * 0.5 = 5 cm. - Finding PR: Similarly, LN (12 cm) corresponds to PR. Thus, PR can be computed as:
PR = LN * k = 12 cm * 0.5 = 6 cm.
Examples & Analogies
Imagine a photograph of a person standing at a distance and a closer portrait of the same person. The full-length photograph represents triangle LMN while the closer portrait represents triangle PQR. The scale of the photograph (how much smaller it is compared to the actual size) is the same as the scale factor you've calculated. Just like the portrait captures the same proportions but appears smaller, the triangles represent similar shapes where every part corresponds to one another.
Problem 2: Proportional Shadows
Chapter 2 of 2
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Chapter Content
A vertical pole 2 meters tall casts a shadow 3 meters long. At the same time, a nearby building casts a shadow 18 meters long. How tall is the building? (Hint: The sun's rays create similar triangles).
Detailed Explanation
This problem uses the concept of similar triangles in real life. The vertical pole and the building, along with their shadows, form two triangles that are similar to each other due to having the same angle of the sunlight hitting them.
1. Set Up Ratios: You can set up a ratio from the heights and shadows because the triangles are similar. The ratio of the height of the pole to its shadow will equal the ratio of the height of the building to its shadow.
(Height of pole / Length of pole's shadow) = (Height of building / Length of building's shadow)
(2 m / 3 m) = (Height of building / 18 m).
2. Calculate Height of Building: Let h be the height of the building. Thus, you can write:
(2 / 3) = (h / 18).
3. Cross-Multiply to Solve for h:
2 * 18 = 3 * h
36 = 3h
h = 36 / 3 = 12 m.
Therefore, the building is 12 meters tall.
Examples & Analogies
Think about how a gardener measures plants in their garden using the shadows they cast on a sunny day. If a small plant casts a long shadow and a big tree casts a longer shadow, by observing the ratios of their heights and shadow lengths, the gardener can determine the height of the tree simply by knowing the height of the small plant. Itβs like they are comparing similar triangles formed by the sunlight regardless of the real distances.
Key Concepts
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Similarity: Shapes with the same shape but different sizes.
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Scale Factor: The ratio of the dimensions of a similar image to its corresponding shape.
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Proportions: Ratios that are equal, used in calculating unknown lengths in similar shapes.
Examples & Applications
If Triangle ABC is similar to Triangle DEF with scale factor k = 3, then if AB = 2 cm, DE = 6 cm.
If rectangle A has dimensions 4 cm by 6 cm and rectangle B has dimensions 8 cm by 12 cm, they are similar with a scale factor of 2.
Memory Aids
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Rhymes
If the angles match up right, similarity takes flight.
Stories
Imagine two trees that are the same shape but grow at different heights. They represent how shapes can be scaled.
Memory Tools
SAS - Similarity - Angles Same.
Acronyms
SSS
Same Shape
Scaled Size.
Flash Cards
Glossary
- Similarity
Figures that have the same shape but may be different in size.
- Scale Factor
The constant ratio by which all corresponding linear dimensions of a shape are multiplied to obtain the dimensions of a similar image.
- Proportion
An equation stating that two ratios are equal, often used in similarity.
Reference links
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