Question

find the value of x so that the area of the rectangle is equal to the area of the triangle. the rectangle has height 7 and length ( x + 6 ); the triangle has base 14 and height ( 2x - 2 ). ( x = square )

Explanation:

Step1: Calculate area of rectangle

The formula for the area of a rectangle is \( A_{rectangle} = length \times width \). Here, length is \( x + 6 \) and width is \( 7 \), so \( A_{rectangle} = 7(x + 6) \).

Step2: Calculate area of triangle

The formula for the area of a triangle is \( A_{triangle} = \frac{1}{2} \times base \times height \). Here, base is \( 14 \) and height is \( 2x - 2 \), so \( A_{triangle} = \frac{1}{2} \times 14 \times (2x - 2) \). Simplify this: \( \frac{14}{2} \times (2x - 2) = 7(2x - 2) \).

Step3: Set areas equal and solve for x

Set \( 7(x + 6) = 7(2x - 2) \). Divide both sides by \( 7 \): \( x + 6 = 2x - 2 \). Subtract \( x \) from both sides: \( 6 = x - 2 \). Add \( 2 \) to both sides: \( x = 8 \).

Answer:

\( 8 \)