Question

the length of the base of an isosceles triangle is x. the length of a leg is 4x - 3. the perimeter of the triangle is 111. find x.

Explanation:

Step1: Write the perimeter formula

The perimeter \(P\) of an isosceles triangle is \(P=\text{base}+2\times\text{leg}\). Given base \(=x\), leg \( = 4x - 3\), and \(P = 111\), so \(x+2(4x - 3)=111\).

Step2: Expand the equation

Expand \(2(4x - 3)\) using the distributive property \(a(b + c)=ab+ac\). Here \(a = 2\), \(b=4x\), \(c=-3\), so \(2(4x - 3)=8x-6\). The equation becomes \(x + 8x-6=111\).

Step3: Combine like - terms

Combine \(x\) and \(8x\). \(x+8x=9x\), so the equation is \(9x-6 = 111\).

Step4: Isolate the variable term

Add 6 to both sides of the equation. \(9x-6 + 6=111 + 6\), which simplifies to \(9x=117\).

Step5: Solve for \(x\)

Divide both sides by 9. \(x=\frac{117}{9}\).

Answer:

\(x = 13\)