a right-angled triangle with hypotenuse (16), one leg (12), and the other leg (x).

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Pythagorean Theorem", "Square Roots", "Error Analysis" ], "new_concepts": [], "current_concepts": [ "Pythagorean Theorem", "Square Roots", "Error Analysis" ] } ## Identify the given values and target variable $$ \begin{aligned} &\text{Hypotenuse } (c) = 16\ &\text{Leg 1 } (a) = 12\ &\text{Leg 2 } (b) = x \end{aligned} $$ ## Apply the Pythagorean theorem to solve for x $$ \begin{aligned} &a^2 + b^2 = c^2\ &12^2 + x^2 = 16^2\ &144 + x^2 = 256\ &x^2 = 256 - 144\ &x^2 = 112\ &x = \sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7} \approx 10.58 \end{aligned} $$ ## Perform error analysis on the handwritten work $$ \begin{aligned} &\text{Handwritten step: } x^2 = 256 - 144\ &\text{Handwritten subtraction error: } 256 - 144 = 112 \quad (\text{incorrectly calculated as } 122)\ &\text{Handwritten square root error: } \sqrt{122} \approx 11.05 \quad (\text{incorrectly calculated as } 10) \end{aligned} $$ The correct value of $x$ is $4\sqrt{7}$ (or approximately $10.58$). There are two errors in the handwritten work: 1. **Subtraction Error:** $256 - 144$ is $112$, but it was incorrectly calculated as $122$. 2. **Square Root Error:** The square root of $122$ is approximately $11.05$, but it was incorrectly simplified to $10$ (since $\sqrt{100} = 10$). { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "C", "color": "#583C87", "withLabel": false } }, { "type": "point", "params": [[6, 0]], "properties": { "name": "A", "color": "#583C87", "withLabel": false } }, { "type": "point", "params": [[6, 5.29]], "properties": { "name": "B", "color": "#583C87", "withLabel": false } }, { "type": "segment", "params": [[0, 0], [6, 0]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "type": "segment", "params": [[6, 0], [6, 5.29]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3 } }, { "type": "segment", "params": [[0, 0], [6, 5.29]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3 } }, { "type": "text", "params": [[3, -0.5], "12"], "properties": { "color": "#8C55F2" } }, { "type": "text", "params": [[6.3, 2.6], "x"], "properties": { "color": "#8C55F2" } }, { "type": "text", "params": [[2.5, 3.0], "16"], "properties": { "color": "#5583F2" } }, { "type": "polygon", "params": [[5.6, 0], [5.6, 0.4], [6, 0.4], [6, 0]], "properties": { "fillColor": "rgba(140, 85, 242, 0.2)", "strokeColor": "#8C55F2", "strokeWidth": 1 } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Pythagorean Theorem" ] }

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Identify the given values and target variable

Apply the Pythagorean theorem to solve for x

Perform error analysis on the handwritten work

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