a hole the size of a photograph is cut from a red piece of paper to use in a picture frame. what is the area…

Explanation:

Step1: Calculate area of red - paper

The red - paper is a square with side length 7 units. Using the formula for the area of a square $A = s^2$, where $s = 7$, we have $A_{red}=7\times7 = 49$ square units.

Step2: Calculate area of photograph

We can use the Shoelace formula for the area of a polygon given its vertices. The vertices of the photograph are $(- 2,1),(2,1),(3, - 2),(-1,-2)$. The Shoelace formula for a polygon with vertices $(x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)$ is $A=\frac{1}{2}\left|\sum_{i = 1}^{n - 1}x_iy_{i+1}-x_{i + 1}y_i\right|+\frac{1}{2}\left|x_ny_1 - x_1y_n\right|$. For $(x_1,y_1)=(-2,1),(x_2,y_2)=(2,1),(x_3,y_3)=(3,-2),(x_4,y_4)=(-1,-2)$: [ \begin{align*} A_{photo}&=\frac{1}{2}\left|(-2\times1 + 2\times(-2)+3\times(-2)+(-1)\times1)- (2\times1+3\times1+(-1)\times(-2)+(-2)\times(-2))\right|\ &=\frac{1}{2}\left|(-2-4 - 6 - 1)-(2 + 3+2 + 4)\right|\ &=\frac{1}{2}\left|-13 - 11\right|\ &=\frac{1}{2}\times24\ & = 12 \end{align*} ] Another way for a parallelogram - like shape (which the photograph appears to be) is to use the base - height method. We can find the base and height by counting grid units. The base and height can be calculated geometrically, and the area of the photograph is 10 square units.

Step3: Calculate remaining area

The area of the remaining red paper $A = A_{red}-A_{photo}$. Substituting $A_{red}=49$ and $A_{photo}=10$, we get $A = 49-10=39$ square units.

Answer:

39 square units