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Anton Sysoev
Anton Sysoev

Lesson 7 Homework Practice Area And Perimeter Of Similar Figures Answer Key



How to Find the Area and Perimeter of Similar Figures




Similar figures are figures that have the same shape but not necessarily the same size. They have corresponding angles that are congruent and corresponding sides that are proportional. In this article, we will learn how to find the area and perimeter of similar figures using ratios and proportions.




lesson 7 homework practice area and perimeter of similar figures answer key


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Perimeter of Similar Figures




The perimeter of a figure is the total length of its boundary. To find the perimeter of a similar figure, we can use the scale factor, which is the ratio of the lengths of corresponding sides. For example, if two similar figures have a scale factor of 3:2, then their perimeters have the same ratio. That means if one figure has a perimeter of 18 units, then the other figure has a perimeter of 12 units.


To find the perimeter of a similar figure, we can use this formula:


Perimeter of image = Scale factor à Perimeter of preimage


where image is the new figure and preimage is the original figure.


Area of Similar Figures




The area of a figure is the amount of space it covers. To find the area of a similar figure, we can use the square of the scale factor, which is the ratio of the areas of corresponding regions. For example, if two similar figures have a scale factor of 3:2, then their areas have a ratio of 9:4. That means if one figure has an area of 36 square units, then the other figure has an area of 16 square units.


To find the area of a similar figure, we can use this formula:


Area of image = (Scale factor)Â Ã Area of preimage


Example




Let's look at an example problem:


The triangles below are similar. Find their perimeters and areas.


Solution:


We can see that the scale factor is 4:3, since 12/9 = 4/3. To find the perimeters, we multiply the scale factor by the perimeter of the preimage:


Perimeter of smaller triangle = 4 Ã (9 + 12 + 15) = 144 units


Perimeter of larger triangle = 3 Ã (12 + 16 + 20) = 144 units


To find the areas, we square the scale factor and multiply it by the area of the preimage. We can use Heron's formula to find the area of a triangle given its side lengths:


Area = â(s(s - a)(s - b)(s - c))


where s is half the perimeter and a, b, and c are the side lengths.


Area of smaller triangle = (4/3)Â Ã â(18(18 - 9)(18 - 12)(18 - 15))


= (16/9) Ã â(18 Ã 9 Ã 6 Ã 3)


= (16/9) Ã â(5832)


= (16/9) Ã 76.33


= 135.99 square units


Area of larger triangle = (3/4)Â Ã â(24(24 - 12)(24 - 16)(24 - 20))


= (9/16) Ã â(24 Ã 12 Ã 8 Ã 4)


= (9/16) Ã â(9216)


= (9/16) Ã 96


= 54 square units


Similarity Transformations




A similarity transformation is a transformation that preserves the shape of a figure but changes its size. There are three types of similarity transformations: dilation, reflection, and rotation. A dilation is a transformation that enlarges or reduces a figure by a scale factor. A reflection is a transformation that flips a figure over a line of symmetry. A rotation is a transformation that turns a figure around a fixed point. Similarity transformations can be combined to create more complex transformations.


Similarity Criteria for Triangles




There are three ways to prove that two triangles are similar: by angle-angle (AA) criterion, by side-side-side (SSS) criterion, and by side-angle-side (SAS) criterion. The AA criterion states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. The SSS criterion states that if the corresponding sides of two triangles are proportional, then the triangles are similar. The SAS criterion states that if two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.


Applications of Similar Figures




Similar figures can be used to model real-world situations and solve problems involving indirect measurement, scale drawings, maps, and proportions. For example, we can use similar triangles to find the height of a tree, a building, or a mountain without measuring them directly. We can also use similar figures to create scale drawings or models of objects or places that are too large or too small to draw or build at actual size. We can use proportions to find missing measurements or compare ratios of different quantities. c481cea774


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