Question

the coordinates of rhombus abcd are a(-4, -2), b(-2, 6), c(6, 8), and d(4, 0). what is the area of the rhombus? round to the nearest whole number, if necessary. 30 square units 60 square units 102 square units 120 square units

Explanation:

Step1: Recall the area formula for a rhombus

The area of a rhombus $A=\frac{1}{2}d_1d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals.

Step2: Find the length of diagonal $AC$

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $A(-4,-2)$ and $C(6,8)$, we have:
\[

$$\begin{align*} d_{AC}&=\sqrt{(6-(-4))^2+(8 - (-2))^2}\\ &=\sqrt{(6 + 4)^2+(8+ 2)^2}\\ &=\sqrt{10^2+10^2}\\ &=\sqrt{100 + 100}\\ &=\sqrt{200}=10\sqrt{2} \end{align*}$$

\]

Step3: Find the length of diagonal $BD$

For points $B(-2,6)$ and $D(4,0)$, we have:
\[

$$\begin{align*} d_{BD}&=\sqrt{(4-(-2))^2+(0 - 6)^2}\\ &=\sqrt{(4 + 2)^2+(0 - 6)^2}\\ &=\sqrt{6^2+(-6)^2}\\ &=\sqrt{36+36}\\ &=\sqrt{72}=6\sqrt{2} \end{align*}$$

\]

Step4: Calculate the area of the rhombus

Substitute $d_1 = 10\sqrt{2}$ and $d_2=6\sqrt{2}$ into the area formula $A=\frac{1}{2}d_1d_2$.
\[

$$\begin{align*} A&=\frac{1}{2}\times10\sqrt{2}\times6\sqrt{2}\\ &=\frac{1}{2}\times10\times6\times(\sqrt{2}\times\sqrt{2})\\ &=\frac{1}{2}\times10\times6\times2\\ &=60 \end{align*}$$

\]

Answer:

B. 60 square units