andi is proving that a quadrilateral ( f(-4,-1), g(-2,-5), h(4,-2), i(2,2) ) is a rectangle. she states that the diagonals are congruent. to prove that she is correct, what are the lengths of the diagonals?
a ( sqrt{45} )
b ( sqrt{65} )
c 65
d do not select this choice
e 45

Question

Accepted Answer

# Explanation:
## Step1: Calculate the length of diagonal $FH$
Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $F(-4,-1)$ and $H(4,-2)$, $x_1=-4,y_1 = - 1,x_2=4,y_2=-2$.
$$
\begin{align*}
d_{FH}&=\sqrt{(4-(-4))^2+(-2 - (-1))^2}\
&=\sqrt{(4 + 4)^2+(-2 + 1)^2}\
&=\sqrt{8^2+(-1)^2}\
&=\sqrt{64 + 1}\
&=\sqrt{65}
\end{align*}
$$
## Step2: Calculate the length of diagonal $GI$
For points $G(-2,-5)$ and $I(2,2)$, $x_1=-2,y_1=-5,x_2=2,y_2 = 2$.
$$
\begin{align*}
d_{GI}&=\sqrt{(2-(-2))^2+(2-(-5))^2}\
&=\sqrt{(2 + 2)^2+(2 + 5)^2}\
&=\sqrt{4^2+7^2}\
&=\sqrt{16+49}\
&=\sqrt{65}
\end{align*}
$$

# Answer:
B. $\sqrt{65}$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the length of diagonal \(FH\)

Step2: Calculate the length of diagonal \(GI\)

Answer: