The length of the sides of a triangle is p, q, r respectively. If $p^2 + q^2 + r^2$ = pq + qr + pr, then triangle is :

Question

Accepted Answer

To determine the type of triangle based on the given condition $ p^2 + q^2 + r^2 = pq + qr + pr $, we need to analyze the properties of the triangle that satisfy this equation.

### Explanation for Option 2: Equilateral Triangle

1. **Understanding the Equation**: The equation $ p^2 + q^2 + r^2 = pq + qr + pr $ can be rewritten as:
   $$
   p^2 + q^2 + r^2 - pq - qr - pr = 0
   $$
   This equation is a specific form that is satisfied when the triangle is equilateral. For an equilateral triangle, all sides are equal, i.e., $ p = q = r $.

2. **Substituting Equal Sides**: If $ p = q = r $, then substituting into the equation gives:
   $$
   p^2 + p^2 + p^2 = p \cdot p + p \cdot p + p \cdot p
   $$
   $$
   3p^2 = 3p^2
   $$
   This is a true statement, confirming that the equation holds for an equilateral triangle.

3. **Conclusion**: Since the equation holds true when all sides are equal, it confirms that the triangle is equilateral. Therefore, Option 2 is correct.

### Explanation for Why Other Options are Incorrect

1. **Option 1: Isosceles Triangle**:
   - An isosceles triangle has at least two equal sides, but not necessarily all three. The equation $ p^2 + q^2 + r^2 = pq + qr + pr $ specifically holds when all three sides are equal, not just two. Therefore, this option is incorrect.

2. **Option 3: Right Angle Triangle**:
   - A right angle triangle satisfies the Pythagorean theorem, $ a^2 + b^2 = c^2 $, where $ c $ is the hypotenuse. The given equation does not conform to this form and does not imply a right angle triangle. Thus, this option is incorrect.

3. **Option 4: Obtuse Angle Triangle**:
   - An obtuse angle triangle has one angle greater than 90 degrees, which affects the relationship between the sides. The given equation does not indicate such a relationship and is not characteristic of an obtuse angle triangle. Therefore, this option is incorrect.

In conclusion, the equation $ p^2 + q^2 + r^2 = pq + qr + pr $ is specifically satisfied by an equilateral triangle, making Option 2 the correct choice.