# The perimeter of triangle ABC is 16.8 meters. What is the side BC if AB refers to BC as 7: 5 BC refers to AC as 3: 4?

According to the conditions of the assignment, AB + BC + AC = 16.8 m.

In addition, the terms of the assignment state that AB: BC = 7: 5 and BC: AC = 3: 4.

Let’s use the main property of proportion. Then, we get: 5 * AB = 7 * BC and 3 * AC = 4 * BC. These equalities allow us to express AB and AC in terms of BC.

We have: AB = (7/5) * BC and AC = (4/3) * BC. Substitute the found expressions into the equality from item 1. Then, we get: (7/5) * BC + BC + (4/3) * BC = 16.8 m or (7/5 + 1 + 4/3) * BC = 16.8 m.

Let’s calculate: 7/5 + 1 + 4/3 = (7 * 3 + 1 * 15 + 4 * 5) / 15 = 56/15.

So, (56/15) * BC = 16.8 m, whence BC = (16.8 m): (56/15) = (15 * 16.8 / 56) m = 4.5 m.

Now let’s find the other sides of the triangle and check the results. We have: AB = (7/5) * 4.5 m = 6.3 m and AC = (4/3) * 4.5 m = 6 m.Let’s find a semi-perimeter (16.8 m): 2 = 8.4 m. Since the semiperimeter is greater than any side of the triangle, then a triangle with such sides exists.

Answer: 4.5 m.