### 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```

Answered: Image /qna-images/answer/839730dc-e42b-4270-b217-c7d2aba6975e.jpg

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

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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