9. find the missing values for the following similar figures.

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Vertical Angles", "Alternate Interior Angles", "Triangle Similarity", "Similar Triangles Application" ], "new_concepts": [], "current_concepts": [ "Vertical Angles", "Alternate Interior Angles", "Triangle Similarity", "Similar Triangles Application" ] } ## Identify similar triangles Using the **Triangle Similarity** and **Alternate Interior Angles** knowledge points $$ \begin{aligned} &\text{The top and bottom horizontal lines are parallel, indicated by the arrows.}\ &\text{The transversal lines create alternate interior angles:}\ &\bullet\ \text{The angle marked with a solid dot on the bottom left is congruent to the solid dot angle on the top right.}\ &\bullet\ \text{The angle marked with an open circle on the bottom is congruent to the open circle angle on the top.}\ &\bullet\ \text{The vertical angles marked with an 'X' at the intersection are congruent.}\ &\text{Thus, the bottom triangle is similar to the top triangle by AA Similarity.} \end{aligned} $$ ## Set up the similarity ratio Using the **Similar Triangles Application** knowledge point $$ \begin{aligned} &\text{Identify corresponding sides opposite to congruent angles:}\ &\bullet\ \text{Opposite to the 'X' angles: sides of length } 10 \text{ (bottom) and } 3 \text{ (top).}\ &\bullet\ \text{Opposite to the open circle angles: sides of length } 15 \text{ (bottom) and } y \text{ (top).}\ &\bullet\ \text{Opposite to the solid dot angles: sides of length } x \text{ (bottom) and } 2 \text{ (top).}\ &\text{This gives the ratio: } \frac{10}{3} = \frac{15}{y} = \frac{x}{2} \end{aligned} $$ ## Solve for x Using the **Similar Triangles Application** knowledge point $$ \begin{aligned} &\frac{x}{2} = \frac{10}{3}\ &x = 2 \cdot \frac{10}{3} = \frac{20}{3} \approx 6.67 \end{aligned} $$ ## Solve for y Using the **Similar Triangles Application** knowledge point $$ \begin{aligned} &\frac{15}{y} = \frac{10}{3}\ &10y = 45\ &y = \frac{45}{10} = 4.5 \end{aligned} $$ $x = \frac{20}{3}$ (or $6.67$) $y = 4.5$ (or $\frac{9}{2}$) { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Similar Triangles Application" ] }

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Question

Explanation:

Identify similar triangles

Set up the similarity ratio

Solve for x

Solve for y

Answer: