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 $14\text{ m}$ more than the distance between $a$ and $c$, and the distance between $b$ and $c$ is $14\text{ m}$ 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": [ "Pythagorean Theorem Applications", "Solving Quadratic Equations" ], "new_concepts": [], "current_concepts": [ "Pythagorean Theorem Applications", "Solving Quadratic Equations", "Quadratic Word Problems" ] } ## Define variables for the distances Using the **Pythagorean Theorem Applications** knowledge point Let the distance between mirror $A$ and mirror $C$ be $x$ meters. - Distance between $A$ and $B$: $x + 14$ - Distance between $B$ and $C$: $x - 14$ ## Set up the geometric equation Using the **Pythagorean Theorem Applications** knowledge point Since the mirrors form a right triangle, the hypotenuse is the longest side, $x + 14$. $$ (x - 14)^2 + x^2 = (x + 14)^2 $$ ## Expand and simplify the equation Using the **Solving Quadratic Equations** knowledge point $$ \begin{aligned} (x^2 - 28x + 196) + x^2 &= x^2 + 28x + 196\ 2x^2 - 28x + 196 &= x^2 + 28x + 196\ x^2 - 56x &= 0 \end{aligned} $$ ## Solve for the distance x Using the **Solving Quadratic Equations** knowledge point $$ \begin{aligned} x(x - 56) &= 0\ x = 56 \quad &\text{or} \quad x = 0 \end{aligned} $$ Since distance must be positive, $x = 56$. 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 $14\text{ m}$ more than the distance between $A$ and $C$, and the distance between $B$ and $C$ is $14\text{ m}$ less than the distance between $A$ and $C$, what is the distance between mirror $A$ and mirror $C$? $56$ m { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "C", "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[5.6, 0]], "properties": { "name": "A", "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, 4.2]], "properties": { "name": "B", "color": "#8C55F2", "withLabel": true } }, { "type": "segment", "params": [[0, 0], [5.6, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3, "name": "x", "withLabel": true, "labelPosition": [2.8, -0.3] } }, { "type": "segment", "params": [[0, 0], [0, 4.2]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3, "name": "x - 14", "withLabel": true, "labelPosition": [-0.8, 2.1] } }, { "type": "segment", "params": [[5.6, 0], [0, 4.2]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 3, "name": "x + 14", "withLabel": true, "labelPosition": [3.0, 2.3] } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Pythagorean Theorem Applications" ] }

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Question

Explanation:

Define variables for the distances

Set up the geometric equation

Expand and simplify the equation

Solve for the distance x

Answer: