14. two segments, $ \overline{ab} $ and $ \overline{cd} $, have endpoints at $ a(-5,11) $, $ b(3,5) $, $ c(9,3) $, and $ d(2,10) $. which of the two segments is longer? provide evidence to support your answer.

Question

Accepted Answer

# Explanation:
## Step1: Recall the distance formula
The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

## Step2: Calculate the length of $\overline{AB}$
For points $A(-5,11)$ and $B(3,5)$, we have $x_1=-5,y_1 = 11,x_2=3,y_2 = 5$.
Substitute into the distance formula:
$$
\begin{align*}
AB&=\sqrt{(3 - (-5))^2+(5 - 11)^2}\
&=\sqrt{(3 + 5)^2+(- 6)^2}\
&=\sqrt{8^2+(-6)^2}\
&=\sqrt{64 + 36}\
&=\sqrt{100}\
& = 10
\end{align*}
$$

## Step3: Calculate the length of $\overline{CD}$
For points $C(9,3)$ and $D(2,10)$, we have $x_1 = 9,y_1=3,x_2 = 2,y_2=10$.
Substitute into the distance formula:
$$
\begin{align*}
CD&=\sqrt{(2 - 9)^2+(10 - 3)^2}\
&=\sqrt{(-7)^2+7^2}\
&=\sqrt{49+49}\
&=\sqrt{98}\
&\approx9.899
\end{align*}
$$

## Step4: Compare the lengths
Since $10>9.899$, we can see that $AB>CD$.

# Answer:
The segment $\overline{AB}$ is longer. The length of $\overline{AB}$ is $10$ and the length of $\overline{CD}$ is $\sqrt{98}\approx9.899$, and $10>\sqrt{98}$.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the distance formula

Step2: Calculate the length of \(\overline{AB}\)

Step3: Calculate the length of \(\overline{CD}\)

Step4: Compare the lengths

Answer: