If a, b, c are the sides of a triangle ABC, then \({a^{\frac{1}{p}}} + {b^{\frac{1}{p}}} - {c^{\frac{1}{p}}}\) where p > 1, is
always positive
Concept:
- The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
Calculation:
Given: a, b, c are the sides of a triangle ABC
Since a, b, c are positive numbers.
⇒ a1/p, b1/p, c1/p are positive numbers. ...(∵ p > 1)
We know that the sum of any two sides must be greater than the third side.
⇒ a1/p + b1/p > c1/p
⇒ a1/p + b1/p - c1/p > 0
∴ Option 2 is correct.
Similar Questions
Consider the following statements:
1. If ABC is a right-angled triangle, right-angled at A, and if sin \(\rm B = \frac 1 3,\) then cosec C = 3.
2. If b cos B = c cos C and if the triangle ABC is not right-angled, then ABC must be isosceles.
Which of the above statements is/are correct?
- In a triangle \(ABC\), if \(a\), \(b\) and \(c\) are the lengths of the sides opposite to the angles \(A\), \(B\) and \(C\) respectively, then what is \(\frac{\sin(A-B)}{\sin(A+B)}\) equal to ?
The sides of a triangle are m, n and \(\rm \sqrt{m^2+n^2+mn}\) . What is the sum of the acute angles of the triangle?
The sides of the triangle are in
sin A, sin B, sin C are in
What is the value of n ?
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Consider the following statements :
1. ABC is right angled triangle
2. The angles of the triangle are in AP
Which of the statements given above is/are correct ?
If c = 8, what is the area of the triangle ?
What is the value of a + b + √2 c equal to ?
Important Questions from Properties of Triangles
Consider the following statements:
1. If ABC is a right-angled triangle, right-angled at A, and if sin \(\rm B = \frac 1 3,\) then cosec C = 3.
2. If b cos B = c cos C and if the triangle ABC is not right-angled, then ABC must be isosceles.
Which of the above statements is/are correct?
- In a triangle \(ABC\), if \(a\), \(b\) and \(c\) are the lengths of the sides opposite to the angles \(A\), \(B\) and \(C\) respectively, then what is \(\frac{\sin(A-B)}{\sin(A+B)}\) equal to ?
In equilateral ΔABC, D and E are points on the sides AB and AC, respectively, such that AD = CE. BE and CD intersect at F. The measure (in degrees) of ∠CFB is:
In Δ ABC, ∠B = 90º, AC = 29 cm and BC = 20 cm. Then \(\frac{{1 - \sin A + \cos A}}{{1 + \sin A + \cos A}}\) is equal to:
The sides of a triangle are m, n and \(\rm \sqrt{m^2+n^2+mn}\) . What is the sum of the acute angles of the triangle?
