Example Question - triangle ratios
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Solving Triangle Ratios
The image shows two triangles with side lengths provided. The larger triangle has sides of length 21, 15, and 18, while the smaller triangle has one side labeled as 5 and two sides that are not labeled. It is stated that the ratio of the larger triangle to the smaller triangle is 3. To solve the problem, we'll assume that the corresponding sides of the triangles are proportional, meaning that the larger triangle is a scaled-up version of the smaller one. Since the ratio is 3, it implies that every length in the larger triangle is 3 times the length of the corresponding length in the smaller triangle. The smaller triangle has a side with a length of 5. To find the corresponding side on the larger triangle, we can divide one of the side lengths of the larger triangle by 3. Let's choose the side length of 15 for simplicity: 15 (larger side length) / 5 (smaller side length) = 3 This confirms that the side length of 5 in the smaller triangle corresponds to the side length of 15 in the larger triangle. To maintain the same scale, other sides of the smaller triangle should be 1/3 of their respective sides in the larger triangle. Thus, the other two sides of the smaller triangle can be found by dividing the remaining sides of the larger triangle by 3: 21 / 3 = 7 18 / 3 = 6 So the smaller triangle should have side lengths of 5, 7, and 6. However, since your question doesn't specify what needs to be solved, but only provides the ratio and the sizes of the triangles, I've provided you with the calculation that confirms the given ratio and the side lengths of the smaller triangle based on that ratio. If there's a specific question or calculation you need beyond this, please provide that information.
Determining Similarity of Triangles by Side Ratios
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 / 3 For the right triangle: Left side to base side = 15 m / 20 m = 3 / 4 Since 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 = 8/15 For the right triangle: Right side to left side = 4.5 m / 15 m = 4.5/15 = 3/10 Since 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 = 8/22.5 For the right triangle: Right side to base side = 4.5 m / 20 m = 4.5/20 = 9/40 The 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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