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Which of the following statements cannot be true?
I. If the length of the sides of a right angled triangle are in arithmetic progression, the length of one of its sides can be 1809 units.
II. A primitive Pythagorean triplet is one where the GCD of the three numbers in the triplet is 1. If an integer greater than 25 belongs to a Pythagorean triplet, then that triplet is non-primitive.
III. The area of a right angled triangle will always be an odd integer.  
  • a)
    I only
  • b)
    II only
  • c)
    II and III
  • d)
    I and II
Correct answer is option 'C'. Can you explain this answer?
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Which of the following statements cannot be true?I. If the length of t...
Solution: Statement I is correct: If (a, b, c) is a Pythagorean triplet then (ka, kb, kc) is also a Pythagorean triplet for any natural number k. (3, 4, 5) is a Pythagorean triplet. (3k, 4k, 5k) will also be a Pythagorean triplet for any natural number k.
(1809, 2412, 3015) is a Pythagorean triplet (for k = 603) and 1809, 2412 and 3015 are in A.P.
Statement II is incorrect: (28, 45, 53) is a primitive Pythagorean triplet with length of all the sides greater than 25.
Statement III is incorrect: For the right angled triangle with length of sides 3, 4 and 5, its area = 1/2 x 3 x 4 = 6 square units, which is even. Moreover, the area of a right triangle need not always be an integer value.
Hence, option 3.
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Which of the following statements cannot be true?I. If the length of t...
False Statements in Geometry

I. If the length of the sides of a right angled triangle are in arithmetic progression, the length of one of its sides can be 1809 units.

II. A primitive Pythagorean triplet is one where the GCD of the three numbers in the triplet is 1. If an integer greater than 25 belongs to a Pythagorean triplet, then that triplet is non-primitive.

III. The area of a right angled triangle will always be an odd integer.

Explanation:

I. This statement cannot be true because if the sides of a right-angled triangle are in an arithmetic progression, the largest side must be the hypotenuse, which is always the longest side. Therefore, the length of one of its sides cannot be 1809 units.

II. This statement is true. A Pythagorean triplet is a set of three positive integers a, b, and c that satisfy the equation a² + b² = c². If an integer greater than 25 belongs to a Pythagorean triplet, then it can be expressed as a multiple of a primitive Pythagorean triplet. Since the GCD of the numbers in a primitive Pythagorean triplet is 1, any multiple of a primitive triplet will have a GCD greater than 1, making it non-primitive.

III. This statement cannot be true because the area of a right-angled triangle can be either an even or odd integer depending on the length of its legs. For example, a right-angled triangle with legs of length 3 and 4 has an area of 6, which is an even integer.

In conclusion, statement III is the false statement as the area of a right-angled triangle can be either an even or odd integer.
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Which of the following statements cannot be true?I. If the length of the sides of a right angled triangle are in arithmetic progression, the length of one of its sides can be 1809 units.II. A primitive Pythagorean triplet is one where the GCD of the three numbers in the triplet is 1. If an integer greater than 25 belongs to a Pythagorean triplet, then that triplet is non-primitive.III. The area of a right angled triangle will always be an odd integer. a)I onlyb)II onlyc)II and IIId)Iand IICorrect answer is option 'C'. Can you explain this answer?
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