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Triangles A and B are similar triangles, as pictured below. Q The smaller triangle has a perimeter of 21 ft and an area of 16 ft². If the larger triangle has a perimeter of 42 ft, what will it's area be? ft2

Triangles A and B are similar triangles, as pictured below. Q The smaller triangle has a perimeter of 21 ft and an area of 16 ft². If the larger triangle has a perimeter of 42 ft, what will it's area be? ft2

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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### Understanding Similar Triangles: An Applied Example #### Problem Statement: We have two triangles, Triangle A and Triangle B, which are similar to each other. This means that the angles of Triangle A are congruent to the angles of Triangle B, and the sides of Triangle A are proportional to the sides of Triangle B. #### Visual Representation: The image below depicts the two similar triangles. - The smaller triangle (Triangle A) is shown on the left. - The larger triangle (Triangle B) is shown on the right. ![Similar Triangles]( path/to/image) #### Given Information: - The perimeter of the smaller triangle (Triangle A) is 21 ft. - The area of the smaller triangle (Triangle A) is 16 ft². - The perimeter of the larger triangle (Triangle B) is 42 ft. #### Objective: We need to find the area of the larger triangle (Triangle B). #### Detailed Solution: 1. **Understanding Perimeter Ratios in Similar Triangles:** - Since the triangles are similar, their corresponding sides are proportional. - The ratio of the perimeters will be the same as the ratio of the corresponding sides. \[ \frac{\text{Perimeter of Triangle B}}{\text{Perimeter of Triangle A}} = \frac{42 \text{ ft}}{21 \text{ ft}} = 2 \] Therefore, each side of Triangle B is twice the length of the corresponding side of Triangle A. 2. **Scaling the Area:** - The area of similar triangles scales by the square of the ratio of their corresponding sides. - Since the ratio of the sides of Triangle B to Triangle A is 2, the ratio of their areas will be \(2^2 = 4\). \[ \text{Area of Triangle B} = \text{Area of Triangle A} \times \left(\frac{\text{Side of Triangle B}}{\text{Side of Triangle A}}\right)^2 = 16 \text{ ft}^2 \times 4 = 64 \text{ ft}^2 \] #### Conclusion: The area of the larger triangle (Triangle B) is \(64 \text{ ft}^2\). Feel free to enter the result below: ```markdown [_________________] ft² ```
Transcribed Image Text:### Understanding Similar Triangles: An Applied Example #### Problem Statement: We have two triangles, Triangle A and Triangle B, which are similar to each other. This means that the angles of Triangle A are congruent to the angles of Triangle B, and the sides of Triangle A are proportional to the sides of Triangle B. #### Visual Representation: The image below depicts the two similar triangles. - The smaller triangle (Triangle A) is shown on the left. - The larger triangle (Triangle B) is shown on the right. ![Similar Triangles]( path/to/image) #### Given Information: - The perimeter of the smaller triangle (Triangle A) is 21 ft. - The area of the smaller triangle (Triangle A) is 16 ft². - The perimeter of the larger triangle (Triangle B) is 42 ft. #### Objective: We need to find the area of the larger triangle (Triangle B). #### Detailed Solution: 1. **Understanding Perimeter Ratios in Similar Triangles:** - Since the triangles are similar, their corresponding sides are proportional. - The ratio of the perimeters will be the same as the ratio of the corresponding sides. \[ \frac{\text{Perimeter of Triangle B}}{\text{Perimeter of Triangle A}} = \frac{42 \text{ ft}}{21 \text{ ft}} = 2 \] Therefore, each side of Triangle B is twice the length of the corresponding side of Triangle A. 2. **Scaling the Area:** - The area of similar triangles scales by the square of the ratio of their corresponding sides. - Since the ratio of the sides of Triangle B to Triangle A is 2, the ratio of their areas will be \(2^2 = 4\). \[ \text{Area of Triangle B} = \text{Area of Triangle A} \times \left(\frac{\text{Side of Triangle B}}{\text{Side of Triangle A}}\right)^2 = 16 \text{ ft}^2 \times 4 = 64 \text{ ft}^2 \] #### Conclusion: The area of the larger triangle (Triangle B) is \(64 \text{ ft}^2\). Feel free to enter the result below: ```markdown [_________________] ft² ```
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