https://file.aiquickdraw.com/mathbot/file/1726894257739hvlswzt8.jpg

The problem depicted in the image asks whether the triangle \( \triangle ABC \) is a right triangle, based on the Pythagorean theorem \( (AB)^2 + (BC)^2 = (AC)^2 \).

Let’s go through each option:

1. **Option A**:
   - Triangle has sides \( AB = 3 \), \( BC = 4 \), and \( AC = 5 \).
   - According to the Pythagorean theorem:
     \[
     (AB)^2 + (BC)^2 = (AC)^2 \quad \Rightarrow \quad 3^2 + 4^2 = 5^2
     \]
     \[
     9 + 16 = 25 \quad \text{which is true.}
     \]
     Therefore, option A is correct: this is a right triangle.

2. **Option B**:
   - Triangle has sides \( AB = 6 \), \( BC = 8 \), and \( AC = 10 \).
   - Check using the Pythagorean theorem:
     \[
     (AB)^2 + (BC)^2 = (AC)^2 \quad \Rightarrow \quad 6^2 + 8^2 = 10^2
     \]
     \[
     36 + 64 = 100 \quad \text{which is true.}
     \]
     However, the statement says that it is false. This is a contradiction.

3. **Option C**:
   - Triangle has sides \( AB = 3 \), \( BC = 4 \), and \( AC = 5 \), but it’s arranged incorrectly, and this does not form a triangle that satisfies the Pythagorean theorem. The assumption is false as these sides do not form a triangle in this orientation.

**Correct answer:**
- Option **A** is true (and should be marked as true), while options **B** and **C** are false for the reasons mentioned.

Would you like more details on any specific part of this solution? Here are 5 related questions to deepen your understanding:

1. How does the Pythagorean theorem confirm if a triangle is right-angled?
2. Why is option B marked as false, even though the Pythagorean relation holds?
3. Can you explain what conditions need to be met for three sides to form a valid triangle?
4. Is it possible for a triangle to have the same side lengths but not be right-angled?
5. How would you test whether any random triangle is a right triangle?

**Tip:** In a right triangle, the square of the length of the hypotenuse is always equal to the sum of the squares of the other two sides.

If in ∆ABC, (AB)^2 + (BC)^2 = (AC)^2, then ∆ABC is a right triangle. Determine if the statements are True or False for each triangle given.

This problem demonstrates how to determine if a triangle is a right triangle using the Pythagorean theorem. Given triangle sides, we verify if (AB)^2 + (BC)^2 = (AC)^2. Suitable for students in Grades 6-8.

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