Question

solve for x. the polygon in each pair are similar.

Explanation:

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<reasoning>

Identify corresponding sides

The problem states that the polygons are similar.
We identify the corresponding sides of the two similar quadrilaterals:

• The left side of length \(12\) corresponds to the left side of length \(30\).
• The bottom side of length \(10\) corresponds to the bottom side of length \(2x + 3\).

Set up the proportion

Since the polygons are similar, the ratio of their corresponding sides must be equal:
\[
\frac{12}{30} = \frac{10}{2x + 3}
\]

Simplify the ratio

Simplify the fraction on the left side:
\[
\frac{12}{30} = \frac{2}{5}
\]
So the equation becomes:
\[
\frac{2}{5} = \frac{10}{2x + 3}
\]

Solve for x

Cross-multiply to solve the equation:
\[
2(2x + 3) = 5 \times 10
\]
\[
4x + 6 = 50
\]
\[
4x = 44
\]
\[
x = 11
\]
</reasoning>

<answer>
<mcq-option>9</mcq-option>
<mcq-correct>11</mcq-correct>
<mcq-option>6</mcq-option>
<mcq-option>5</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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Answer:

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</pre_analysis>

<reasoning>

Identify corresponding sides

The problem states that the polygons are similar.
We identify the corresponding sides of the two similar quadrilaterals:

• The left side of length \(12\) corresponds to the left side of length \(30\).
• The bottom side of length \(10\) corresponds to the bottom side of length \(2x + 3\).

Set up the proportion

Since the polygons are similar, the ratio of their corresponding sides must be equal:
\[
\frac{12}{30} = \frac{10}{2x + 3}
\]

Simplify the ratio

Simplify the fraction on the left side:
\[
\frac{12}{30} = \frac{2}{5}
\]
So the equation becomes:
\[
\frac{2}{5} = \frac{10}{2x + 3}
\]

Solve for x

Cross-multiply to solve the equation:
\[
2(2x + 3) = 5 \times 10
\]
\[
4x + 6 = 50
\]
\[
4x = 44
\]
\[
x = 11
\]
</reasoning>

<answer>
<mcq-option>9</mcq-option>
<mcq-correct>11</mcq-correct>
<mcq-option>6</mcq-option>
<mcq-option>5</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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"Mathematics",
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