Question

A triangle has one side length of 7a + 3 and a second side length of 8b - 2a. If the perimeter of the triangle is 12a + 7b + 5c, what is the length of the third side?

Answer 1

Let x be the length of the third side of the triangle.

7a + 3 + 8b - 2a + x = 12a + 7b + 5c

5a + 8b - 3 + x = 12a + 7b + 5c

x = 7a - b + 5c - 3

Answer 2

To find the length of the third side of the triangle with a perimeter of 12a + 7b + 5c and two known sides being 7a + 3 and 8b - 2a, we subtract the sum of the known sides from the perimeter. The resulting length of the third side is 7a - b + 5c - 3.

Step-by-step explanation:

Given that a triangle has one side length of 7a + 3 and a second side length of 8b - 2a, we can find the length of the third side by using the fact that the perimeter of a triangle is the sum of the lengths of all its sides.

The perimeter of the given triangle is 12a + 7b + 5c. We can set up an equation to solve for the length of the third side (let's call it 'side c'):

(7a + 3) + (8b - 2a) + side c = 12a + 7b + 5c

To find 'side c', we'll consolidate like terms and solve the equation:

1. Combine the terms with 'a': 7a - 2a = 5a
2. Add the 'a' and 'b' terms to the left side: 5a + 8b
3. Add the constant terms: 3 to the left side
4. Subtract the sum of these terms from the given perimeter: (12a + 7b + 5c) - (5a + 8b + 3)
5. Solve for 'side c': 'side c' = 12a + 7b + 5c - 5a - 8b - 3
6. Simplify the equation: 'side c' = (12a - 5a) + (7b - 8b) + 5c - 3
7. Calculate the final expression: 'side c' = 7a - b + 5c - 3

Therefore, the length of the third side of the triangle is 7a - b + 5c - 3.