Question

in the diagram, \\(ab = 10\\) and \\(ac = 2\sqrt{10}\\). what is the perimeter of \\(\triangle abc\\)?

\\(\bigcirc\\) 10 units
\\(\bigcirc\\) \\(10 + 2\sqrt{10}\\) units
\\(\bigcirc\\) 20 units
\\(\bigcirc\\) \\(20 + 2\sqrt{10}\\) units

Explanation:

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<reasoning>

Identify the given side lengths

Using the Perimeter in Coordinate Plane knowledge point
\[

$$\begin{aligned} &AB = 10\\ &AC = 2\sqrt{10} \end{aligned}$$

\]

Calculate the length of the third side BC

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &B = (-5, -2),\quad C = (5, -2)\\ &BC = \sqrt{(5 - (-5))^2 + (-2 - (-2))^2} = \sqrt{10^2 + 0^2} = 10 \end{aligned}$$

\]

Sum the side lengths to find the perimeter

Using the Perimeter in Coordinate Plane knowledge point
\[

$$\begin{aligned} &\text{Perimeter} = AB + AC + BC\\ &\text{Perimeter} = 10 + 2\sqrt{10} + 10 = 20 + 2\sqrt{10} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>10 units</mcq-option>
<mcq-option>10 + 2\(\sqrt{10}\) units</mcq-option>
<mcq-option>20 units</mcq-option>
<mcq-correct>20 + 2\(\sqrt{10}\) units</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify the given side lengths

Using the Perimeter in Coordinate Plane knowledge point
\[

$$\begin{aligned} &AB = 10\\ &AC = 2\sqrt{10} \end{aligned}$$

\]

Calculate the length of the third side BC

Using the Pythagorean Theorem knowledge point
\[

$$\begin{aligned} &B = (-5, -2),\quad C = (5, -2)\\ &BC = \sqrt{(5 - (-5))^2 + (-2 - (-2))^2} = \sqrt{10^2 + 0^2} = 10 \end{aligned}$$

\]

Sum the side lengths to find the perimeter

Using the Perimeter in Coordinate Plane knowledge point
\[

$$\begin{aligned} &\text{Perimeter} = AB + AC + BC\\ &\text{Perimeter} = 10 + 2\sqrt{10} + 10 = 20 + 2\sqrt{10} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>10 units</mcq-option>
<mcq-option>10 + 2\(\sqrt{10}\) units</mcq-option>
<mcq-option>20 units</mcq-option>
<mcq-correct>20 + 2\(\sqrt{10}\) units</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
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