Question

geometry write and solve an equation to find the value of x so that the figures have the same area.
area of rectangle = (length)(width)
area of triangle = 1/2(base)(height)
○ 3(x + 6) = 2(8)(x + 4) - 34
○ 3(x + 6) = 1/2(8)(x + 4) - 2
○ 3(x + 4) = 1/2(8)(x + 6) - 12
○ 3(x + 6) = 1/2(x + 4) - 36

Explanation:

Step1: Calculate the area of the triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\). Here, the base is \(8\) and the height is \((x + 4)\). So, the area of the triangle \(A_{triangle}=\frac{1}{2}\times8\times(x + 4)=4(x + 4)=4x+16\).

Step2: Calculate the area of the rectangle

The formula for the area of a rectangle is \(A = length\times width\). Here, the length is \((x + 6)\) and the width is \(3\). So, the area of the rectangle \(A_{rectangle}=3(x + 6)=3x + 18\).

Step3: Set the areas equal and solve for \(x\)

Since the areas are equal, we set \(4x+16=3x + 18\).
Subtract \(3x\) from both sides: \(4x-3x+16=3x-3x + 18\), which gives \(x+16=18\).
Subtract \(16\) from both sides: \(x+16 - 16=18-16\), so \(x = 2\).

Answer:

\(x = 2\)