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If in an equilateral triangle, inradius is a rational number then which of the following is not true.
- a)circumradius is always rational
- b)Exradii are always rational
- c)Area is always irrational
- d)Perimeter is always rational
Correct answer is option 'D'. Can you explain this answer?
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Explanation:
Given: In an equilateral triangle, the inradius is a rational number.
Circumradius:
- The circumradius of an equilateral triangle can be calculated using the formula R = 2r, where R is the circumradius and r is the inradius.
- Since the inradius is given to be rational, the circumradius will also be rational (as it is twice the inradius).
- Therefore, the statement that the circumradius is always rational is true.
Exradii:
- Exradii are the radii of the three excircles of the triangle.
- The exradii are related to the inradius by the formula r1 = r2 = r3 = r, where r1, r2, r3 are the exradii and r is the inradius.
- Since the inradius is rational, the exradii will also be rational.
- Therefore, the statement that the exradii are always rational is true.
Area:
- The area of an equilateral triangle can be calculated using the formula A = (s^2 * √3) / 4, where A is the area and s is the side length of the triangle.
- Since the side length of an equilateral triangle is directly related to the inradius (s = 3r), and the inradius is rational, the side length will also be rational.
- However, the area formula involves the square root of 3, which is an irrational number.
- Therefore, the statement that the area is always irrational is true.
Perimeter:
- The perimeter of an equilateral triangle is simply 3 times the side length, which, as mentioned earlier, is rational.
- Therefore, the perimeter of the equilateral triangle will always be rational.
- The statement that the perimeter is always rational is not true.
Therefore, the correct answer is option 'D', as the perimeter of the equilateral triangle is not always rational when the inradius is a rational number.
Given: In an equilateral triangle, the inradius is a rational number.
Circumradius:
- The circumradius of an equilateral triangle can be calculated using the formula R = 2r, where R is the circumradius and r is the inradius.
- Since the inradius is given to be rational, the circumradius will also be rational (as it is twice the inradius).
- Therefore, the statement that the circumradius is always rational is true.
Exradii:
- Exradii are the radii of the three excircles of the triangle.
- The exradii are related to the inradius by the formula r1 = r2 = r3 = r, where r1, r2, r3 are the exradii and r is the inradius.
- Since the inradius is rational, the exradii will also be rational.
- Therefore, the statement that the exradii are always rational is true.
Area:
- The area of an equilateral triangle can be calculated using the formula A = (s^2 * √3) / 4, where A is the area and s is the side length of the triangle.
- Since the side length of an equilateral triangle is directly related to the inradius (s = 3r), and the inradius is rational, the side length will also be rational.
- However, the area formula involves the square root of 3, which is an irrational number.
- Therefore, the statement that the area is always irrational is true.
Perimeter:
- The perimeter of an equilateral triangle is simply 3 times the side length, which, as mentioned earlier, is rational.
- Therefore, the perimeter of the equilateral triangle will always be rational.
- The statement that the perimeter is always rational is not true.
Therefore, the correct answer is option 'D', as the perimeter of the equilateral triangle is not always rational when the inradius is a rational number.
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If in an equilateral triangle, inradius is a rational number then which of the following is not true.a)circumradius is always rationalb)Exradii are always rationalc)Area is always irrationald)Perimeter is always rationalCorrect answer is option 'D'. Can you explain this answer?
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