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Sum of the lengths of any two sides of a triangle is greater than the length of the ____.
- a)first side
- b)second side
- c)third side
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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Sum of the lengths of any two sides of a triangle is greater than the ...
We have Triangle Inequality Theorem which states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. This means that as you know the two sides greater than the third, you know what all sides will not make a triangle.
Most Upvoted Answer
Sum of the lengths of any two sides of a triangle is greater than the ...
Hi friend ,here is your answerNo! it's not necessa
ry. The sum of any 2 angles of a triangle, may be greater or shorter than
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Sum of the lengths of any two sides of a triangle is greater than the ...
Explanation:
In a triangle, the sum of the lengths of any two sides is always greater than the length of the third side. This is known as the Triangle Inequality Theorem.
Triangle Inequality Theorem:
The Triangle Inequality Theorem states that for any triangle with sides of lengths a, b, and c, the following inequality holds true:
a + b > c
b + c > a
a + c > b
Proof:
To understand why this theorem is true, let's consider a simple example.
Suppose we have a triangle with sides of lengths 3, 4, and 5. We can test the theorem by substituting the values into the inequality:
3 + 4 > 5
7 > 5
This inequality is true, so the theorem holds for this example.
General Case:
Now let's consider the general case. Suppose we have a triangle with sides of lengths a, b, and c. We can assume without loss of generality that a ≤ b ≤ c.
To prove the Triangle Inequality Theorem, we need to show that:
a + b > c
b + c > a
a + c > b
1. a + b > c:
We can assume a ≤ b ≤ c. Since c is the longest side, a + b < 2c.="" therefore,="" a="" +="" b="" /> c.
2. b + c > a:
Since a ≤ b ≤ c, a + b < 2c.="" therefore,="" b="" +="" c="" /> a.
3. a + c > b:
Since a ≤ b ≤ c, a + b < 2c.="" therefore,="" a="" +="" c="" /> b.
Conclusion:
Therefore, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. Thus, the correct answer is option 'C', which is the third side.
In a triangle, the sum of the lengths of any two sides is always greater than the length of the third side. This is known as the Triangle Inequality Theorem.
Triangle Inequality Theorem:
The Triangle Inequality Theorem states that for any triangle with sides of lengths a, b, and c, the following inequality holds true:
a + b > c
b + c > a
a + c > b
Proof:
To understand why this theorem is true, let's consider a simple example.
Suppose we have a triangle with sides of lengths 3, 4, and 5. We can test the theorem by substituting the values into the inequality:
3 + 4 > 5
7 > 5
This inequality is true, so the theorem holds for this example.
General Case:
Now let's consider the general case. Suppose we have a triangle with sides of lengths a, b, and c. We can assume without loss of generality that a ≤ b ≤ c.
To prove the Triangle Inequality Theorem, we need to show that:
a + b > c
b + c > a
a + c > b
1. a + b > c:
We can assume a ≤ b ≤ c. Since c is the longest side, a + b < 2c.="" therefore,="" a="" +="" b="" /> c.
2. b + c > a:
Since a ≤ b ≤ c, a + b < 2c.="" therefore,="" b="" +="" c="" /> a.
3. a + c > b:
Since a ≤ b ≤ c, a + b < 2c.="" therefore,="" a="" +="" c="" /> b.
Conclusion:
Therefore, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. Thus, the correct answer is option 'C', which is the third side.
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Sum of the lengths of any two sides of a triangle is greater than the length of the ____.a)first sideb)second sidec)third sided)none of theseCorrect answer is option 'C'. Can you explain this answer?
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