# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-7/13

Exercises: Try Out - Set 7: Real exam simulation to sharpen your skills and build confidence. - Problem 13

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## Exercise 13

### Question

import { Graph } from "../graph";

export const metadata = {
  title: "Problem 13",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

<Graph
  title="Triangle ABC"
  description={
    <span>
      Visualization of Triangle <InlineMath math="ABC" /> with points <InlineMath math="P, Q," /> and <InlineMath math="R" />.
    </span>
  }
/>

Triangle <InlineMath math="ABC" /> is an isosceles triangle with <InlineMath math="AC = BC" />. Determine the measure of angle <InlineMath math="\angle BPC" />!

Decide whether statements <InlineMath math="(1)" /> and <InlineMath math="(2)" /> below are sufficient to answer the question!

1. <InlineMath math="\angle CAB = 40^\circ" />.
2. <InlineMath math="PC" /> is the angle bisector of triangle <InlineMath math="ABC" />.


### Choices

- [ ] Statement $$(1)$$ ALONE is sufficient, but statement $$(2)$$ ALONE is not sufficient.
- [x] Statement $$(2)$$ ALONE is sufficient, but statement $$(1)$$ ALONE is not sufficient.
- [ ] BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- [ ] EACH statement ALONE is sufficient.
- [ ] Statements $$(1)$$ and $$(2)$$ TOGETHER are NOT sufficient.

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 13",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

We are asked to determine the measure of angle <InlineMath math="\angle BPC" /> in an isosceles triangle <InlineMath math="ABC" /> with <InlineMath math="AC = BC" />.

#### Analysis of Statement 1

Given <InlineMath math="\angle CAB = 40^\circ" />. Since <InlineMath math="\triangle ABC" /> is isosceles with <InlineMath math="AC = BC" />, the base angles are equal:

<MathContainer>
  <BlockMath math="\angle CBA = \angle CAB = 40^\circ" />
</MathContainer>

We can calculate the vertex angle <InlineMath math="\angle ACB" />:

<MathContainer>
  <BlockMath math="\angle ACB = 180^\circ - (40^\circ + 40^\circ) = 100^\circ" />
</MathContainer>

However, this information does not specify the position of point <InlineMath math="P" /> on side <InlineMath math="AB" />. Point <InlineMath math="P" /> can be anywhere along the segment <InlineMath math="AB" />. Consequently, the measure of <InlineMath math="\angle BPC" /> varies depending on the position of <InlineMath math="P" />. Therefore, statement <InlineMath math="(1)" /> ALONE is not sufficient to answer the question.

#### Analysis of Statement 2

Given <InlineMath math="PC" /> is the angle bisector of triangle <InlineMath math="ABC" />. In this context, <InlineMath math="PC" /> is the bisector of angle <InlineMath math="C" /> intersecting side <InlineMath math="AB" /> at point <InlineMath math="P" />.

In an isosceles triangle with <InlineMath math="AC = BC" />, the angle bisector drawn from the vertex (the angle between the equal sides) to the base has special properties. It is also:

1. An altitude (perpendicular to the base).
2. A median (bisects the base).

Since <InlineMath math="PC" /> is an altitude, <InlineMath math="PC \perp AB" />. Thus, the angle formed is a right angle:

<MathContainer>
  <BlockMath math="\angle BPC = 90^\circ" />
</MathContainer>

We obtain a definite value for <InlineMath math="\angle BPC" />. Therefore, statement <InlineMath math="(2)" /> ALONE is sufficient to answer the question.

#### Conclusion

Statement <InlineMath math="(1)" /> is not sufficient, while statement <InlineMath math="(2)" /> is sufficient. Thus, **statement <InlineMath math="(2)" /> ALONE is sufficient, but statement <InlineMath math="(1)" /> ALONE is not sufficient**.


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