Question

3 - 47 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 calculate the length of each side of the triangle. be sure to confirm that your answer is correct.

Explanation:

Step1: Set up the equation for the perimeter

The perimeter of a triangle is the sum of its side lengths. So, \((7x - 4)+(10x + 3)+19 = 52\).
Simplify the left - hand side: \(7x-4 + 10x+3 + 19=(7x + 10x)+(-4 + 3+19)=17x+18\).
The equation becomes \(17x + 18=52\).

Step2: Solve for \(x\)

Subtract 18 from both sides of the equation: \(17x+18 - 18=52 - 18\), which gives \(17x=34\).
Divide both sides by 17: \(x=\frac{34}{17}=2\).

Step3: Calculate the side lengths

For the side \(7x - 4\): Substitute \(x = 2\), \(7\times2-4=14 - 4 = 10\).
For the side \(10x + 3\): Substitute \(x = 2\), \(10\times2+3=20 + 3=23\).
The third side is 19.

Step4: Verify the perimeter

Check the sum of the side lengths: \(10 + 23+19=52\).

Answer:

\(x = 2\), the side lengths are 10 units, 23 units, and 19 units.