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$\triangle A B C$ has side lengths of 10 units, 20 units, and 24 units. $\triangle X Y Z$ is similar to $\triangle A B C$, and the length of its longest side is 60 units.
The perimeter of $\triangle X Y Z$ is [BLANK] units. If the height of $\triangle A B C$, with respect to its longest side being the base, is 8 units, the area of $\triangle X Y Z$ is [BLANK] square units.
Answer (2)
Calculate the perimeter of △ A BC : P A BC = 10 + 20 + 24 = 54 .
Determine the ratio of corresponding sides: k = 24 60 = 2.5 .
Calculate the perimeter of △ X Y Z : P X Y Z = 2.5 × 54 = 135 .
Calculate the area of △ X Y Z : A X Y Z = 2 1 × 60 × ( 2.5 × 8 ) = 600 .
The perimeter of △ X Y Z is 135 units and the area of △ X Y Z is 600 square units.
Explanation
Analyze the problem and given data We are given two triangles, △ A BC and △ X Y Z . We know the side lengths of △ A BC are 10, 20, and 24 units. We also know that △ X Y Z is similar to △ A BC , and its longest side is 60 units. The height of △ A BC with respect to its longest side is 8 units. We need to find the perimeter and area of △ X Y Z .
Calculate the perimeter of triangle ABC First, let's find the perimeter of △ A BC . The perimeter is the sum of the lengths of its sides: P A BC = 10 + 20 + 24 = 54 So, the perimeter of △ A BC is 54 units.
Determine the ratio of corresponding sides Next, we need to find the ratio of corresponding sides between the two similar triangles. The longest side of △ X Y Z is 60 units, and the longest side of △ A BC is 24 units. Therefore, the ratio k is: k = 24 60 = 2 5 = 2.5 This means that each side of △ X Y Z is 2.5 times the length of the corresponding side of △ A BC .
Calculate the perimeter of triangle XYZ Now, we can find the perimeter of △ X Y Z by multiplying the perimeter of △ A BC by the ratio k :
P X Y Z = k × P A BC = 2.5 × 54 = 135 So, the perimeter of △ X Y Z is 135 units.
Calculate the height of triangle XYZ We are given that the height of △ A BC with respect to its longest side (24 units) is 8 units. Since △ X Y Z is similar to △ A BC , the height of △ X Y Z with respect to its longest side will also be k times the height of △ A BC . Therefore, the height of △ X Y Z is: h X Y Z = k × h A BC = 2.5 × 8 = 20 So, the height of △ X Y Z is 20 units.
Calculate the area of triangle XYZ Finally, we can calculate the area of △ X Y Z . The area of a triangle is given by the formula: A = 2 1 × ba se × h e i g h t In △ X Y Z , the base is the longest side, which is 60 units, and the height is 20 units. Therefore, the area of △ X Y Z is: A X Y Z = 2 1 × 60 × 20 = 600 So, the area of △ X Y Z is 600 square units.
State the final answer The perimeter of △ X Y Z is 135 units, and the area of △ X Y Z is 600 square units.
Examples
Understanding similar triangles is useful in many real-world applications, such as architecture and engineering. For example, when creating scale models of buildings or bridges, the principles of similarity ensure that the proportions are maintained, and the model accurately represents the real structure. This allows architects and engineers to test designs and identify potential issues before construction begins, saving time and resources.
The perimeter of triangle XYZ is 135 units, and the area is 600 square units.
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