6.rectangle qrst
q(-4,-1)
r(1,4)
s(4,1)
t(-1,-4)
perimeter
area

Question

Accepted Answer

# Explanation:
## Step1: Calculate the length of side QR
Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $Q(-4,-1)$ and $R(1,4)$, $x_1=-4,y_1 = - 1,x_2=1,y_2 = 4$.
$$
\begin{align*}
QR&=\sqrt{(1-(-4))^2+(4 - (-1))^2}\
&=\sqrt{(1 + 4)^2+(4+1)^2}\
&=\sqrt{5^2+5^2}\
&=\sqrt{25 + 25}\
&=\sqrt{50}=5\sqrt{2}
\end{align*}
$$
## Step2: Calculate the length of side RS
For points $R(1,4)$ and $S(4,1)$, $x_1 = 1,y_1=4,x_2 = 4,y_2=1$
$$
\begin{align*}
RS&=\sqrt{(4 - 1)^2+(1 - 4)^2}\
&=\sqrt{3^2+(-3)^2}\
&=\sqrt{9+9}\
&=\sqrt{18}=3\sqrt{2}
\end{align*}
$$
## Step3: Calculate the perimeter of the rectangle
The perimeter formula of a rectangle $P=2(l + w)$, where $l = 5\sqrt{2}$, $w=3\sqrt{2}$
$$
\begin{align*}
P&=2(5\sqrt{2}+3\sqrt{2})\
&=2\times8\sqrt{2}\
&=16\sqrt{2}
\end{align*}
$$
## Step4: Calculate the area of the rectangle
The area formula of a rectangle $A=l\times w$, where $l = 5\sqrt{2}$, $w = 3\sqrt{2}$
$$
\begin{align*}
A&=(5\sqrt{2})\times(3\sqrt{2})\
&=5\times3\times\sqrt{2}\times\sqrt{2}\
&=15\times2\
&=30
\end{align*}
$$

# Answer:
Perimeter: $16\sqrt{2}$, Area: $30$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the length of side QR

Step2: Calculate the length of side RS

Step3: Calculate the perimeter of the rectangle

Step4: Calculate the area of the rectangle

Answer: