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

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

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Pythagorean Theorem", "Perimeter in Coordinate Plane" ], "new_concepts": [], "current_concepts": [ "Pythagorean Theorem", "Perimeter in Coordinate Plane" ] } ## 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 $$ \begin{aligned} &\text{Perimeter} = AB + AC + BC\ &\text{Perimeter} = 10 + 2\sqrt{10} + 10 = 20 + 2\sqrt{10} \end{aligned} $$

10 units

10 + 2$\sqrt{10}$ units

20 units

20 + 2$\sqrt{10}$ units

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Perimeter in Coordinate Plane" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given side lengths

Calculate the length of the third side BC

Sum the side lengths to find the perimeter

Answer: