Question

1. the perimeter of the triangle at right is 52 units. write and solve an equation based on the information in the diagram. use your solution for x to find the measures of each side of the triangle. be sure to confirm that your answer is correct.

Explanation:

Step1: Recall perimeter formula

The perimeter of a triangle is the sum of its three sides. So, we add the three side lengths: \((7x - 4)+(10x + 3)+19 = 52\).

Step2: Simplify the left - hand side

Combine like terms: \(7x+10x-4 + 3+19=52\), which simplifies to \(17x+18 = 52\).

Step3: Solve for x

Subtract 18 from both sides: \(17x=52 - 18\), so \(17x = 34\). Then divide both sides by 17: \(x=\frac{34}{17}=2\).

Step4: Find each side length

• For the side \(7x - 4\): Substitute \(x = 2\), we get \(7(2)-4=14 - 4 = 10\).
• For the side \(10x + 3\): Substitute \(x = 2\), we get \(10(2)+3=20 + 3 = 23\).
• The third side is 19 (given).

Step5: Verify the perimeter

Sum the three sides: \(10+23 + 19=52\), which matches the given perimeter.

Answer:

The value of \(x\) is \(2\). The side lengths of the triangle are \(10\) units, \(23\) units, and \(19\) units.