question 26

using the drawing below, solve for $x$:

$x = $ ________ (blank 1)

Question

question 26

using the drawing below, solve for $x$:

$x = $ ________ (blank 1)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Triangle Similarity", "AA Similarity Postulate" ], "new_concepts": [ "Triangle Proportionality Theorem" ], "current_concepts": [ "Triangle Similarity", "AA Similarity Postulate", "Triangle Proportionality Theorem" ] } ## Identify parallel lines and similar triangles Using the *AA Similarity Postulate* and *Triangle Similarity* knowledge points $$ \begin{aligned} &DE \parallel BC \implies \angle ADE = \angle ABC \text{ and } \angle AED = \angle ACB\ &\triangle ADE \sim \triangle ABC \end{aligned} $$ ## Set up the side length ratios Using the *Triangle Similarity* knowledge point $$ \begin{aligned} &\frac{AD}{AB} = \frac{AE}{AC}\ &AD = 12,\quad AB = AD + DB = 12 + 20 = 32\ &AE = x,\quad AC = AE + EC = x + 15 \end{aligned} $$ ## Solve the proportion for x Using the *Triangle Similarity* knowledge point $$ \begin{aligned} &\frac{12}{32} = \frac{x}{x + 15}\ &\frac{3}{8} = \frac{x}{x + 15}\ &3(x + 15) = 8x\ &3x + 45 = 8x\ &5x = 45\ &x = 9 \end{aligned} $$ Using the drawing below, solve for $x$: $x =$ 9 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Triangle Proportionality Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify parallel lines and similar triangles

Set up the side length ratios

Solve the proportion for x

Answer: