triangle abc has the following vertices: • a(7, -2) • b(-1, -8) • c(5, 0) is triangle abc an isosceles triangle, and why? choose 1 answer: a yes, because ab = bc. b yes, because ac = bc. c no, because $\overline{ac}$ is longer than $\overline{ab}$. d no, because $\triangle abc$ doesnt have a pair of sides with equal lengths.

Question

Accepted Answer

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

## Step2: Calculate the length of $AB$
For points $A(7,-2)$ and $B(-1,-8)$:
$$
\begin{align*}
AB&=\sqrt{(-1 - 7)^2+(-8 - (-2))^2}\
&=\sqrt{(-8)^2+(-6)^2}\
&=\sqrt{64 + 36}\
&=\sqrt{100}\
& = 10
\end{align*}
$$

## Step3: Calculate the length of $BC$
For points $B(-1,-8)$ and $C(5,0)$:
$$
\begin{align*}
BC&=\sqrt{(5 - (-1))^2+(0 - (-8))^2}\
&=\sqrt{(6)^2+(8)^2}\
&=\sqrt{36+64}\
&=\sqrt{100}\
& = 10
\end{align*}
$$

## Step4: Calculate the length of $AC$
For points $A(7,-2)$ and $C(5,0)$:
$$
\begin{align*}
AC&=\sqrt{(5 - 7)^2+(0 - (-2))^2}\
&=\sqrt{(-2)^2+(2)^2}\
&=\sqrt{4 + 4}\
&=\sqrt{8}\
& = 2\sqrt{2}
\end{align*}
$$

## Step5: Compare the lengths
We found that $AB = 10$ and $BC=10$, so $AB = BC$. An isosceles triangle has at least two equal - length sides. Since $AB = BC$, triangle $ABC$ is isosceles.

# Answer:
A. Yes, because $AB = BC$.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the distance formula

Step2: Calculate the length of \(AB\)

Step3: Calculate the length of \(BC\)

Step4: Calculate the length of \(AC\)

Step5: Compare the lengths

Answer: