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 P1P_1 and P2P_2, 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 a1a2=k\frac{a_1}{a_2} = k, then the ratio of the areas is A1A2=k2\frac{A_1}{A_2} = k^2.

Given:

  • A1=45A_1 = 45 (area of the smaller triangle)
  • A2=80A_2 = 80 (area of the larger triangle)

The ratio of the areas is:

A1A2=4580=916\frac{A_1}{A_2} = \frac{45}{80} = \frac{9}{16}

Thus, the ratio of the side lengths kk is:

k=916=34k = \sqrt{\frac{9}{16}} = \frac{3}{4}

This means that the sides of the smaller triangle are 34\frac{3}{4} times the sides of the larger triangle.

Step 2: Express the perimeters in terms of kk

Let the perimeter of the larger triangle be P2P_2. Then the perimeter of the smaller triangle is P1=34P2P_1 = \frac{3}{4}P_2.

We are given that the sum of the perimeters is 35:

P1+P2=35P_1 + P_2 = 35

Substitute P1=34P2P_1 = \frac{3}{4}P_2 into the equation:

34P2+P2=35\frac{3}{4}P_2 + P_2 = 35

Step 3: Solve for P2P_2

Combine the terms:

34P2+44P2=74P2=35\frac{3}{4}P_2 + \frac{4}{4}P_2 = \frac{7}{4}P_2 = 35

Multiply both sides by 4:

7P2=1407P_2 = 140

Now, divide by 7:

P2=20P_2 = 20

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:

  1. How can we find the side lengths of the triangles if we know their areas and perimeters?
  2. What other properties of similar triangles can be derived from the area ratio?
  3. How does the perimeter ratio relate to the side length ratio in similar triangles?
  4. If the sum of the perimeters was different, how would that affect the result?
  5. 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