Question

triangle abc is an isosceles triangle in which side ab = ac. what is the perimeter of triangle abc?

\\(5 + \sqrt{10}\\) units
\\(10 + \sqrt{10}\\) units
\\(10\sqrt{10}\\) units
\\(50\\) units

Explanation:

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<reasoning>

Identify the coordinates of the vertices

Using the Distance Formula and Perimeter in Coordinate Plane knowledge points
\[

$$\begin{aligned} &A = (-4, -2)\\ &B = (-1, 2)\\ &C = (-4, 3) \end{aligned}$$

\]

Calculate the length of side AC

Using the Distance Formula knowledge point
\[

$$\begin{aligned} d(A, C) &= \sqrt{(-4 - (-4))^2 + (3 - (-2))^2}\\ &= \sqrt{0^2 + 5^2}\\ &= 5 \end{aligned}$$

\]

Calculate the length of side AB

Using the Distance Formula and Pythagorean Theorem knowledge points
\[

$$\begin{aligned} d(A, B) &= \sqrt{(-1 - (-4))^2 + (2 - (-2))^2}\\ &= \sqrt{3^2 + 4^2}\\ &= \sqrt{9 + 16}\\ &= 5 \end{aligned}$$

\]

Calculate the length of side BC

Using the Distance Formula and Pythagorean Theorem knowledge points
\[

$$\begin{aligned} d(B, C) &= \sqrt{(-4 - (-1))^2 + (3 - 2)^2}\\ &= \sqrt{(-3)^2 + 1^2}\\ &= \sqrt{9 + 1}\\ &= \sqrt{10} \end{aligned}$$

\]

Compute the perimeter of triangle ABC

Using the Perimeter in Coordinate Plane knowledge point
\[

$$\begin{aligned} \text{Perimeter} &= AB + AC + BC\\ &= 5 + 5 + \sqrt{10}\\ &= 10 + \sqrt{10} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(5 + \sqrt{10}\) units</mcq-option>
<mcq-correct>(B) \(10 + \sqrt{10}\) units</mcq-correct>
<mcq-option>(C) \(10\sqrt{10}\) units</mcq-option>
<mcq-option>(D) \(50\) units</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the coordinates of the vertices

Using the Distance Formula and Perimeter in Coordinate Plane knowledge points
\[

$$\begin{aligned} &A = (-4, -2)\\ &B = (-1, 2)\\ &C = (-4, 3) \end{aligned}$$

\]

Calculate the length of side AC

Using the Distance Formula knowledge point
\[

$$\begin{aligned} d(A, C) &= \sqrt{(-4 - (-4))^2 + (3 - (-2))^2}\\ &= \sqrt{0^2 + 5^2}\\ &= 5 \end{aligned}$$

\]

Calculate the length of side AB

Using the Distance Formula and Pythagorean Theorem knowledge points
\[

$$\begin{aligned} d(A, B) &= \sqrt{(-1 - (-4))^2 + (2 - (-2))^2}\\ &= \sqrt{3^2 + 4^2}\\ &= \sqrt{9 + 16}\\ &= 5 \end{aligned}$$

\]

Calculate the length of side BC

Using the Distance Formula and Pythagorean Theorem knowledge points
\[

$$\begin{aligned} d(B, C) &= \sqrt{(-4 - (-1))^2 + (3 - 2)^2}\\ &= \sqrt{(-3)^2 + 1^2}\\ &= \sqrt{9 + 1}\\ &= \sqrt{10} \end{aligned}$$

\]

Compute the perimeter of triangle ABC

Using the Perimeter in Coordinate Plane knowledge point
\[

$$\begin{aligned} \text{Perimeter} &= AB + AC + BC\\ &= 5 + 5 + \sqrt{10}\\ &= 10 + \sqrt{10} \end{aligned}$$

\]
</reasoning>

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<mcq-option>(A) \(5 + \sqrt{10}\) units</mcq-option>
<mcq-correct>(B) \(10 + \sqrt{10}\) units</mcq-correct>
<mcq-option>(C) \(10\sqrt{10}\) units</mcq-option>
<mcq-option>(D) \(50\) units</mcq-option>
</answer>

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