the points k(7,1), l(0,7), m(-9,9), and n(-2,3) form rhombus klmn. plot the points then click the \graph quadrilateral\ button. then find the area of the rhombus. click on the graph to plot a point. click a point to delete it.

Question

Accepted Answer

# Answer:
$78$

# Explanation:
## Step1: Find the length of diagonal $KM$
Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $K(7,1)$ and $M(-9,9)$, $x_1 = 7,y_1 = 1,x_2=-9,y_2 = 9$.
$$
\begin{align*}
KM&=\sqrt{(-9 - 7)^2+(9 - 1)^2}\
&=\sqrt{(-16)^2+8^2}\
&=\sqrt{256 + 64}\
&=\sqrt{320}\
&=8\sqrt{5}
\end{align*}
$$
## Step2: Find the length of diagonal $LN$
For $L(0,7)$ and $N(-2,3)$, $x_1 = 0,y_1 = 7,x_2=-2,y_2 = 3$.
$$
\begin{align*}
LN&=\sqrt{(-2 - 0)^2+(3 - 7)^2}\
&=\sqrt{(-2)^2+(-4)^2}\
&=\sqrt{4 + 16}\
&=\sqrt{20}\
&=2\sqrt{5}
\end{align*}
$$
## Step3: Calculate the area of the rhombus
The area formula of a rhombus is $A=\frac{1}{2}d_1d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals.
Let $d_1 = KM$ and $d_2 = LN$.
$$
\begin{align*}
A&=\frac{1}{2}\times8\sqrt{5}\times\frac{39}{2\sqrt{5}}\
&=\frac{1}{2}\times(8\sqrt{5}\times\frac{39}{2\sqrt{5}})\
&=\frac{1}{2}\times78\
& = 78
\end{align*}
$$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Find the length of diagonal \(KM\)

Step2: Find the length of diagonal \(LN\)

Step3: Calculate the area of the rhombus

Answer: