Question

1. geometry write and solve an equation to find the value of x so that the figures have the same area.

Explanation:

Step1: Write area formula for triangle

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

Step2: Write area formula for rectangle

The area formula for a rectangle is $A_{rectangle}=length\times width$. Here, length = $x + 6$ and width = 3, so $A_{rectangle}=3(x + 6)$.

Step3: Set areas equal and solve for x

Set $A_{triangle}=A_{rectangle}$, we get the equation $4(x + 4)=3(x + 6)$.
Expand both sides: $4x+16 = 3x + 18$.
Subtract $3x$ from both sides: $4x-3x+16=3x-3x + 18$, which simplifies to $x+16 = 18$.
Subtract 16 from both sides: $x+16-16=18 - 16$.
So $x = 2$.

Answer:

$x = 2$