Math Problem Statement
The areas of two similar triangles are 45 and 80. The sum of their perimeters is 35. Find the perimeter of the larger triangle.
Solution
To find the perimeter of the larger triangle given the areas of two similar triangles and the sum of their perimeters, follow these steps:
Step 1: Establish the relationship between the areas and the side lengths
Let the perimeters of the smaller and larger triangles be and , respectively. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding side lengths. So, if the ratio of the side lengths is , then the ratio of the areas is .
Given:
- (area of the smaller triangle)
- (area of the larger triangle)
The ratio of the areas is:
Thus, the ratio of the side lengths is:
This means that the sides of the smaller triangle are times the sides of the larger triangle.
Step 2: Express the perimeters in terms of
Let the perimeter of the larger triangle be . Then the perimeter of the smaller triangle is .
We are given that the sum of the perimeters is 35:
Substitute into the equation:
Step 3: Solve for
Combine the terms:
Multiply both sides by 4:
Now, divide by 7:
So, the perimeter of the larger triangle is 20 units.
Would you like more details or have any questions? Here are some related questions you might find interesting:
- How can we find the side lengths of the triangles if we know their areas and perimeters?
- What other properties of similar triangles can be derived from the area ratio?
- How does the perimeter ratio relate to the side length ratio in similar triangles?
- If the sum of the perimeters was different, how would that affect the result?
- Can this method be extended to find the perimeters of non-similar triangles?
Tip: When working with similar figures, always remember that the ratio of their areas is the square of the ratio of their corresponding sides.
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Math Problem Analysis
Mathematical Concepts
Similar Triangles
Ratio of Areas
Perimeter
Formulas
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Theorems
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Suitable Grade Level
Grades 7-9