identify a method to find the area of $ \triangle qrs $ given that $ \overline{q s} \perp \overline{r t} $. then find the area. show your work
o a) use the distance formula to find the base $ q s $ and the height $ r t $. $ q s=\sqrt{18}, r t=\sqrt{72} $. then use the area formula:
$ a=\frac{1}{2}(\sqrt{18})(\sqrt{72})=\frac{1}{2}(36) $. so, the area is 18 units².
o b) use the distance formula to find the base $ q s $ and the height $ r t $. $ q s=\sqrt{72}, r t=\sqrt{18} $. then use the area formula:
$ a=\frac{1}{2}(\sqrt{72})(\sqrt{18})=\frac{1}{2}(36) $. so, the area is 18 units².
o c) use the distance formula to find the base $ q s $ and the height $ r t $. $ q s=\sqrt{64}, r t=\sqrt{25} $. then use the area formula:
$ a=\frac{1}{2}(\sqrt{64})(\sqrt{25})=\frac{1}{2}(40) $. so, the area is 20 units².

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Accepted Answer

# Explanation:
## Step1: Calculate the length of $QS$
Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $Q(2,4)$ and $S(8,-2)$, we have $x_1 = 2,y_1=4,x_2 = 8,y_2=-2$.
$$
\begin{align*}
QS&=\sqrt{(8 - 2)^2+(-2 - 4)^2}\
&=\sqrt{6^2+(-6)^2}\
&=\sqrt{36 + 36}\
&=\sqrt{72}
\end{align*}
$$
## Step2: Calculate the length of $RT$
Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $R(6,6)$ and $T(3,3)$, we have $x_1 = 6,y_1=6,x_2 = 3,y_2=3$.
$$
\begin{align*}
RT&=\sqrt{(3 - 6)^2+(3 - 6)^2}\
&=\sqrt{(-3)^2+(-3)^2}\
&=\sqrt{9+9}\
&=\sqrt{18}
\end{align*}
$$
## Step3: Calculate the area of $\triangle QRS$
Since $QS\perp RT$, the area of $\triangle QRS$ is $A=\frac{1}{2}\times base\times height$. Here, base $=QS=\sqrt{72}$ and height $=RT=\sqrt{18}$
$$
\begin{align*}
A&=\frac{1}{2}\times\sqrt{72}\times\sqrt{18}\
&=\frac{1}{2}\times\sqrt{72\times18}\
&=\frac{1}{2}\times\sqrt{9\times8\times9\times2}\
&=\frac{1}{2}\times9\times\sqrt{16}\
&=\frac{1}{2}\times9\times4\
& = 18
\end{align*}
$$

# Answer:
A. Use the Distance Formula to find the base $QS$ and the height $RT$. $QS=\sqrt{72}$, $RT=\sqrt{18}$. Then use the area formula $A=\frac{1}{2}(\sqrt{72})(\sqrt{18})=\frac{1}{2}(36)=18$ units².

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Question

Explanation:

Step1: Calculate the length of \(QS\)

Step2: Calculate the length of \(RT\)

Step3: Calculate the area of \(\triangle QRS\)

Answer: