Question

for the triangles to be similar by the sss similarity theorem, what must be the value of \\(y\\)?

18
20
24
26

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [],
"current_concepts": [
"Triangle Similarity",
"SSS Similarity Theorem"
]
}
</pre_analysis>

<reasoning>

Set up the ratio of corresponding sides

\[
\frac{PR}{AC} = \frac{28}{14} = 2
\]

Relate the remaining corresponding sides

\[

$$\begin{aligned} \frac{PQ}{AB} &= \frac{y}{10} = 2 \\ \frac{RQ}{CB} &= \frac{y}{10} = 2 \end{aligned}$$

\]

Solve for y

\[
y = 10 \times 2 = 20
\]
</reasoning>

<answer>
<mcq-option>18</mcq-option>
<mcq-correct>20</mcq-correct>
<mcq-option>24</mcq-option>
<mcq-option>26</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"SSS Similarity Theorem"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [],
"current_concepts": [
"Triangle Similarity",
"SSS Similarity Theorem"
]
}
</pre_analysis>

<reasoning>

Set up the ratio of corresponding sides

\[
\frac{PR}{AC} = \frac{28}{14} = 2
\]

Relate the remaining corresponding sides

\[

$$\begin{aligned} \frac{PQ}{AB} &= \frac{y}{10} = 2 \\ \frac{RQ}{CB} &= \frac{y}{10} = 2 \end{aligned}$$

\]

Solve for y

\[
y = 10 \times 2 = 20
\]
</reasoning>

<answer>
<mcq-option>18</mcq-option>
<mcq-correct>20</mcq-correct>
<mcq-option>24</mcq-option>
<mcq-option>26</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"SSS Similarity Theorem"
]
}
</post_analysis>