bill uses mirrors to augment the \laser experience\ at a laser show. at one show he places three mirrors, $a, b, c$, in a right triangular form. if the distance between $a$ and $b$ is $11\text{ m}$ more than the distance between $a$ and $c$, and the distance between $b$ and $c$ is $11\text{ m}$ less than the distance between $a$ and $c$, what is the distance between mirror $a$ and mirror $c$?

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Solving Quadratic Equations", "Geometric Quadratic Modeling" ], "new_concepts": [ "Pythagorean Theorem Applications" ], "current_concepts": [ "Solving Quadratic Equations", "Geometric Quadratic Modeling", "Pythagorean Theorem Applications" ] } ## Define variables for the triangle sides Using the **Geometric Quadratic Modeling** knowledge point Let $x$ be the distance between mirror $A$ and mirror $C$ in meters: $$AC = x$$ The distance between $A$ and $B$ is $11\text{ m}$ more than $AC$: $$AB = x + 11$$ The distance between $B$ and $C$ is $11\text{ m}$ less than $AC$: $$BC = x - 11$$ ## Set up the Pythagorean equation Using the **Geometric Quadratic Modeling** knowledge point Since $A, B, C$ form a right triangle, the hypotenuse is the longest side, which is $AB = x + 11$. $$AC^2 + BC^2 = AB^2$$ $$x^2 + (x - 11)^2 = (x + 11)^2$$ ## Expand and simplify the equation Using the **Solving Quadratic Equations** knowledge point $$ \begin{aligned} &x^2 + (x^2 - 22x + 121) = x^2 + 22x + 121\ &2x^2 - 22x + 121 = x^2 + 22x + 121\ &x^2 - 44x = 0 \end{aligned} $$ ## Solve for x Using the **Solving Quadratic Equations** knowledge point $$ \begin{aligned} &x(x - 44) = 0\ &x = 44 \quad (\text{since } x > 11) \end{aligned} $$ Bill uses mirrors to augment the "laser experience" at a laser show. At one show he places three mirrors, $A, B, C$, in a right triangular form. If the distance between $A$ and $B$ is $11\text{ m}$ more than the distance between $A$ and $C$, and the distance between $B$ and $C$ is $11\text{ m}$ less than the distance between $A$ and $C$, what is the distance between mirror $A$ and mirror $C$? 44 m { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "C", "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[4.4, 0]], "properties": { "name": "A", "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, 3.3]], "properties": { "name": "B", "color": "#8C55F2", "withLabel": true } }, { "type": "polygon", "params": [[[0, 0], [4.4, 0], [0, 3.3]]], "properties": { "fillColor": "rgba(140, 85, 242, 0.15)", "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "text", "params": [[2.2, -0.4], "x = 44 m"], "properties": { "color": "#583C87" } }, { "type": "text", "params": [[-1.2, 1.65], "x - 11 = 33 m"], "properties": { "color": "#583C87" } }, { "type": "text", "params": [[2.4, 1.9], "x + 11 = 55 m"], "properties": { "color": "#583C87" } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Geometric Quadratic Modeling" ] }

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Explanation:

Define variables for the triangle sides

Set up the Pythagorean equation

Expand and simplify the equation

Solve for x

Answer: