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If a, b and c are the sides of a triangle, and a2 + b2 + c2 = ab + bc + ca, then the triangle is
- a)equilateral
- b)isosceles
- c)right angled
- d)obtuse angled
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If a, b and c are the sides of a triangle, and a2 + b2 + c2 = ab + bc ...
Given:
- a, b, and c are the sides of a triangle.
- a^2 + b^2 + c^2 = ab + bc + ca.
To prove:
The triangle is equilateral.
1. Introduction:
To prove that the triangle is equilateral, we need to show that all three sides of the triangle are equal.
2. Assumptions:
Let's assume that the triangle is not equilateral, i.e., at least two sides are unequal.
3. Analysis:
- Let's assume a > b > c without loss of generality (since the triangle is not equilateral).
- Now, we have a^2 + b^2 + c^2 = ab + bc + ca. (Given)
- Subtracting 2ab from both sides, we get a^2 - 2ab + b^2 + c^2 = bc + ca - 2ab.
- Simplifying, we get (a - b)^2 + c^2 = c(a - b).
- Dividing both sides by (a - b), we get a - b + c^2 / (a - b) = c.
- Since a > b, a - b > 0. Therefore, c^2 / (a - b) > 0.
- So, a - b + c^2 / (a - b) > 0.
- But c > 0. Therefore, c > a - b + c^2 / (a - b).
4. Inequality:
We have c > a - b + c^2 / (a - b).
- Multiplying both sides by (a - b), we get c(a - b) > (a - b)(a - b + c^2 / (a - b)).
- Simplifying, we get ca - cb > a^2 - ab + bc - b^2 + c^2.
- Rearranging the terms, we get a^2 + b^2 + c^2 > ab + bc + ca. (1)
5. Contradiction:
But, we know that a^2 + b^2 + c^2 = ab + bc + ca. (Given)
- Comparing equations (1) and (Given), we see that they contradict each other.
6. Conclusion:
- Since our assumption (that the triangle is not equilateral) leads to a contradiction, our assumption is false.
- Therefore, the triangle must be equilateral.
7. Final Answer:
The correct answer is option 'A' - the triangle is equilateral.
- a, b, and c are the sides of a triangle.
- a^2 + b^2 + c^2 = ab + bc + ca.
To prove:
The triangle is equilateral.
Proof:
1. Introduction:
To prove that the triangle is equilateral, we need to show that all three sides of the triangle are equal.
2. Assumptions:
Let's assume that the triangle is not equilateral, i.e., at least two sides are unequal.
3. Analysis:
- Let's assume a > b > c without loss of generality (since the triangle is not equilateral).
- Now, we have a^2 + b^2 + c^2 = ab + bc + ca. (Given)
- Subtracting 2ab from both sides, we get a^2 - 2ab + b^2 + c^2 = bc + ca - 2ab.
- Simplifying, we get (a - b)^2 + c^2 = c(a - b).
- Dividing both sides by (a - b), we get a - b + c^2 / (a - b) = c.
- Since a > b, a - b > 0. Therefore, c^2 / (a - b) > 0.
- So, a - b + c^2 / (a - b) > 0.
- But c > 0. Therefore, c > a - b + c^2 / (a - b).
4. Inequality:
We have c > a - b + c^2 / (a - b).
- Multiplying both sides by (a - b), we get c(a - b) > (a - b)(a - b + c^2 / (a - b)).
- Simplifying, we get ca - cb > a^2 - ab + bc - b^2 + c^2.
- Rearranging the terms, we get a^2 + b^2 + c^2 > ab + bc + ca. (1)
5. Contradiction:
But, we know that a^2 + b^2 + c^2 = ab + bc + ca. (Given)
- Comparing equations (1) and (Given), we see that they contradict each other.
6. Conclusion:
- Since our assumption (that the triangle is not equilateral) leads to a contradiction, our assumption is false.
- Therefore, the triangle must be equilateral.
7. Final Answer:
The correct answer is option 'A' - the triangle is equilateral.
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Community Answer
If a, b and c are the sides of a triangle, and a2 + b2 + c2 = ab + bc ...
a2 + b2 + c2 = ab + bc + ac
(a - b)2 + (b - c)2 + (c - a)2 = 0 ...(1)
Since the sum of squares is zero, each term should be zero.
⇒ (a - b)2 = 0, (b - c)2 = 0, (c - a)2 = 0
⇒ a = b = c
Hence, the triangle is equilateral.
(a - b)2 + (b - c)2 + (c - a)2 = 0 ...(1)
Since the sum of squares is zero, each term should be zero.
⇒ (a - b)2 = 0, (b - c)2 = 0, (c - a)2 = 0
⇒ a = b = c
Hence, the triangle is equilateral.
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