Ratio Puzzles with Triangles

If you know the perimeter of a triangle and the ratios of the sides, you can find the lengths of the sides.

Example: The perimeter of a triangle is 84 units. The sides have lengths r, s, and t. The ratio of s to r is 5:3, and the ratio of t to r is 2:1. Find the length of each side.

Since both ratios contain r, rewrite one or both ratios to make r the same. You can write the ratio of t to r as 6:3. Now you can write a three-part ratio.

r:s:t = 3:5:6

There is number x such that r = 3x, s=5x, and t=6x. Since you know the perimeter, 84, you can use algebra to find the lengths of the sides.

r + s + t = 84

3x + 5x + 6x = 84

14x = 84

x = 6

3x = 18, 5x = 30, 6x = 36

So, r = 18, s = 30, and t = 36.

Find the lengths of the sides of each triangle.

1. The perimeter of a triangle is 75 units. The sides have lengths a, b, and c. The ratio of b to a is 3:5, and the ratio of c to a is 7:5. Find the length of each side.

2. The perimeter of a triangle is 88 units. The sides have lengths d, e, and f. The ratio of e to d is 3:1, and the ratio of f to e is 10:9. Find the length of each side.

3. The perimeter of a triangle is 91 units. The sides have lengths p, q, and r. The ratio of p to r is 3:1, and the ratio of q to r is 5:2. Find the length of each side.

4. The perimeter of a triangle is 68 units. The sides have lengths g, h, and j. The ratio of j to g is 2:1, and the ratio of h to g is 5:4. Find the length of each side

5. Write a problem similar to those above involving ratios in triangles.