The sides a,b,c of a triangle are such that a^2+b^2+c^2=ab+bc+ca.What type of triangle is it?
To determine the type of triangle when given the equation a^2+b^2+c^2 = ab+bc+ca, we need to analyze the relationship between the sides of the triangle.
Step 1: Rearrange the equation.
Start by subtracting ab, bc, and ca from both sides of the equation:
a^2 + b^2 + c^2 - ab - bc - ca = 0
Step 2: Simplify.
Rearrange the equation to gather similar terms:
a^2 - ab - ca + b^2 - bc + c^2 = 0
Step 3: Factorize.
The equation can be factored by grouping pairs of terms:
a(a - b - c) + b(b - c) + c(c - a) = 0
Step 4: Analyze the factors.
Based on the factored equation, we can observe the three factors:
1. a(a - b - c)
2. b(b - c)
3. c(c - a)
Step 5: Determine the type of triangle.
By analyzing the factors, we can arrive at the following conclusions:
- If a = 0 or a = b + c, the first factor, a(a - b - c), is zero. This means that one side of the triangle is zero, resulting in a degenerate triangle.
- If a - b - c = 0, the first factor is zero, indicating that the triangle is a right-angled triangle.
- If b - c = 0, it means that b = c. Similarly, if c - a = 0, it means that c = a. Both conditions imply that the triangle is an isosceles triangle.
- If none of the above conditions are met, it means that the triangle is a scalene triangle, where none of the sides are equal.
Therefore, based on the given equation, the triangle can be classified as either a degenerate triangle, a right-angled triangle, an isosceles triangle, or a scalene triangle.