Question - Determining Similarity of Triangles by Side Ratios
Solution:
To determine if the two triangles are similar, we can compare the ratios of corresponding sides. Two triangles are similar if the ratios of the lengths of their corresponding sides are equal.For the triangle on the left in the image, let's label the sides as follows:- The long base side at the bottom is 22.5 m,- The left side is 15 m,- The upper side connecting to the tip of the triangle is 8 m.For the triangle on the right, the sides are:- The long base side at the bottom is 20 m,- The left side is 15 m,- The right side connecting to the tip of the triangle is 4.5 m.Now, let's compare the ratios:For the left triangle:Left side to base side = 15 m / 22.5 m = 2 / 3For the right triangle:Left side to base side = 15 m / 20 m = 3 / 4Since the ratio of the left side to the base side of the left triangle (2/3) is not equal to the ratio of the left side to the base side of the right triangle (3/4), the triangles are not similar.The lengths of the left sides of both triangles, however, are equal to 15 m, suggesting that at least those sides are proportional. To check if the triangles are indeed similar, we should also compare the ratio of the upper side of the left triangle to the right side of the right triangle.For the left triangle:Upper side to left side = 8 m / 15 m = 8/15For the right triangle:Right side to left side = 4.5 m / 15 m = 4.5/15 = 3/10Since the ratio of the upper side to the left side of the left triangle (8/15) is not equal to the ratio of the right side to the left side of the right triangle (3/10), the triangles are confirmed to not be similar based on the sides provided.To be sure, we would also need to check the ratio of the remaining pair of corresponding sides:For the left triangle:Upper side to base side = 8 m / 22.5 m = 8/22.5For the right triangle:Right side to base side = 4.5 m / 20 m = 4.5/20 = 9/40The ratios 8/22.5 and 9/40 are also not equal, further confirming the triangles are not similar.Therefore, based on the ratios of corresponding sides, the two triangles in the image are not similar.
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