# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-5/10

Exercises: Try Out - Set 5: Real exam simulation to sharpen your skills and build confidence. - Problem 10

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## Exercise 10

### Question

export const metadata = {
  title: "Problem 10",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given the following plane figures:

1. Regular hexagon.
2. Isosceles right triangle.
3. Right trapezoid.
4. Non-square rectangle.

How many figures have the same number of lines of symmetry as the order of rotational symmetry?


### Choices

- [ ] $$0$$
- [ ] $$1$$
- [ ] $$2$$
- [x] $$3$$
- [ ] $$4$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 10",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We analyze the lines of symmetry (reflectional symmetry) and rotational symmetry order for each given plane figure:

1. **Regular Hexagon**
   - Lines of symmetry: <InlineMath math="6" /> (lines through vertices and midpoints of sides).
   - Rotational symmetry order: <InlineMath math="6" /> (angles of rotation <InlineMath math="60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ" />).
   - Conclusion: <InlineMath math="6 = 6" />. Figure <InlineMath math="(1)" /> satisfies the condition.

2. **Isosceles Right Triangle**
   - Lines of symmetry: <InlineMath math="1" /> (altitude from the right angle to the hypotenuse).
   - Rotational symmetry order: <InlineMath math="1" /> (only the initial position <InlineMath math="360^\circ" />).
   - Conclusion: <InlineMath math="1 = 1" />. Figure <InlineMath math="(2)" /> satisfies the condition.

3. **Right Trapezoid**
   - Lines of symmetry: <InlineMath math="0" /> (no axis of symmetry).
   - Rotational symmetry order: <InlineMath math="1" /> (only the initial position <InlineMath math="360^\circ" />).
   - Conclusion: <InlineMath math="0 \neq 1" />. Figure <InlineMath math="(3)" /> does not satisfy the condition.

4. **Non-square Rectangle**
   - Lines of symmetry: <InlineMath math="2" /> (vertical and horizontal lines through the center).
   - Rotational symmetry order: <InlineMath math="2" /> (angles of rotation <InlineMath math="180^\circ" /> and <InlineMath math="360^\circ" />).
   - Conclusion: <InlineMath math="2 = 2" />. Figure <InlineMath math="(4)" /> satisfies the condition.

Thus, the figures that have the same number of lines of symmetry as the order of rotational symmetry are figures <InlineMath math="(1)" />, <InlineMath math="(2)" />, and <InlineMath math="(4)" />. There are <InlineMath math="3" /> figures.


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