the circle below has center $k$, and its radius is $5\text{ cm}$. given that $m\angle lkm = 70^\circ$, find the length of the minor arc $\widehat{lm}$.

give an exact answer in terms of $\pi$, and be sure to include the correct unit in your answer.

length of minor arc $\widehat{lm}$ :

Question

the circle below has center $k$, and its radius is $5\text{ cm}$. given that $m\angle lkm = 70^\circ$, find the length of the minor arc $\widehat{lm}$.

give an exact answer in terms of $\pi$, and be sure to include the correct unit in your answer.

length of minor arc $\widehat{lm}$ :

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Central Angle" ], "new_concepts": [ "Arc Length" ], "current_concepts": [ "Central Angle", "Arc Length" ] } ## Identify the given parameters Using the *Central Angle* knowledge point $$ \begin{aligned} &r = 5\text{ cm}\ &\theta = 70^\circ \end{aligned} $$ ## State the arc length formula We need to find the length of the minor arc $ \widehat{LM} $. The formula for the arc length $ s $ of a circle with radius $ r $ and central angle $ \theta $ in degrees is: $$ s = \frac{\theta}{360^\circ} \cdot 2\pi r $$ ## Calculate the exact arc length Substitute the given values into the formula: $$ \begin{aligned} s &= \frac{70}{360} \cdot 2\pi (5) \ s &= \frac{7}{36} \cdot 10\pi \ s &= \frac{70}{36}\pi \ s &= \frac{35}{18}\pi\text{ cm} \end{aligned} $$ The circle below has center K, and its radius is 5 cm. Given that $m\angle LKM = 70^\circ$, find the length of the minor arc $ \widehat{LM} $. Give an exact answer in terms of $\pi$, and be sure to include the correct unit in your answer. Length of minor arc $ \widehat{LM} $: $\frac{35}{18}\pi\text{ cm}$ { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "K", "color": "#583C87", "withLabel": true, "labelPosition": "left" } }, { "type": "circle", "params": [[0, 0], 5], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "point", "params": [[4.698, 1.71]], "properties": { "name": "M", "color": "#583C87", "withLabel": true, "labelPosition": "right" } }, { "type": "point", "params": [[3.214, -3.83]], "properties": { "name": "L", "color": "#583C87", "withLabel": true, "labelPosition": "right" } }, { "type": "segment", "params": [[0, 0], [4.698, 1.71]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[0, 0], [3.214, -3.83]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "text", "params": [0.8, -0.5, "70°"], "properties": { "color": "#583C87" } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Arc Length" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given parameters

State the arc length formula

Calculate the exact arc length

Answer: