The areas of two similar triangles are 45 and 80. The sum of their perimeters is 35. Find the perimeter of the larger triangle.

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 \( P_1 \) and \( P_2 \), 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 \( \frac{a_1}{a_2} = k \), then the ratio of the areas is \( \frac{A_1}{A_2} = k^2 \).

Given:
- \( A_1 = 45 \) (area of the smaller triangle)
- \( A_2 = 80 \) (area of the larger triangle)

The ratio of the areas is:

\[
\frac{A_1}{A_2} = \frac{45}{80} = \frac{9}{16}
\]

Thus, the ratio of the side lengths \( k \) is:

\[
k = \sqrt{\frac{9}{16}} = \frac{3}{4}
\]

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

### Step 2: Express the perimeters in terms of \( k \)
Let the perimeter of the larger triangle be \( P_2 \). Then the perimeter of the smaller triangle is \( P_1 = \frac{3}{4}P_2 \).

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

\[
P_1 + P_2 = 35
\]

Substitute \( P_1 = \frac{3}{4}P_2 \) into the equation:

\[
\frac{3}{4}P_2 + P_2 = 35
\]

### Step 3: Solve for \( P_2 \)
Combine the terms:

\[
\frac{3}{4}P_2 + \frac{4}{4}P_2 = \frac{7}{4}P_2 = 35
\]

Multiply both sides by 4:

\[
7P_2 = 140
\]

Now, divide by 7:

\[
P_2 = 20
\]

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.

Learn how to find the perimeter of a larger triangle when given the areas and perimeters of two similar triangles. Follow step-by-step calculations using the ratio of areas and perimeter relations. Suitable for Grades 7-9.

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