To solve this problem, we are given a triangle with sides represented by algebraic expressions and a known perimeter. We’ll set up and solve an equation based on this information.

Step 1: Define the Given Information

The sides of the triangle are labeled as follows:

1. Side 1: $7x - 4$
2. Side 2: $10x + 3$
3. Side 3: 19

The perimeter of the triangle is 52 units.

Step 2: Set Up the Equation

The perimeter of a triangle is the sum of its three sides. Therefore, we can set up the following equation:

$(7x - 4) + (10x + 3) + 19 = 52$

Step 3: Simplify the Equation

Combine like terms:

$7x + 10x + (-4 + 3 + 19) = 52$ $17x + 18 = 52$

Step 4: Solve for $x$

1. Subtract 18 from both sides: $17x = 34$
2. Divide by 17: $x = 2$

Step 5: Calculate Each Side Length

Now that we have $x = 2$, we can substitute this value back into each expression for the side lengths:

1. Side 1: $7x - 4 = 7(2) - 4 = 14 - 4 = 10$
2. Side 2: $10x + 3 = 10(2) + 3 = 20 + 3 = 23$
3. Side 3: 19 (This side length is given directly)

Step 6: Verify the Perimeter

The calculated side lengths are 10, 23, and 19. Adding these together:

$10 + 23 + 19 = 52$

The perimeter matches the given value, confirming that our solution is correct.

Final Answer

The lengths of the sides of the triangle are:

• Side 1: 10 units
• Side 2: 23 units
• Side 3: 19 units

Would you like further explanation on any part of this process?

Here are five related questions for further practice:

1. What is the perimeter of a triangle if each side length is increased by a constant value?
2. If the perimeter of a triangle is 60 units and two side lengths are given as expressions, how do you solve for the unknown side?
3. How can you set up an equation to solve for $x$ when given the perimeter and side expressions of a triangle?
4. How do you verify the solution to a perimeter problem involving algebraic expressions?
5. What happens to the perimeter of a triangle if one of its sides is halved?

Tip: When solving for $x$ in geometry problems with a known perimeter, double-check each step of your equation setup to ensure accuracy.