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22 Mar 2017, 10:02
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The three sides of triangle have length p, q and r, each an integer. Is this triangle right triangle?
(1) The perimeter of the triangle is an odd integer.
(2) If the triangle's area is doubled, the result is not an integer.
(1) The perimeter of the triangle is an odd integer.
(2) If the triangle's area is doubled, the result is not an integer.
20 May 2017, 10:56
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poojamathur21
The three sides of triangle have length p, q and r, each an integer. Is this triangle right triangle?
(1) The perimeter of the triangle is an odd integer.
Given p + q + r = odd. This implies that either all three p, q and r are odd OR two of them are even and the third one is odd.
If all three p, q and r are odd, then \((side_1)^2 + (side_2)^2 = odd+odd=even\) and \((side_3)^2=odd\). Thus \((side_1)^2 + (side_2)^2 \neq (side_3)^2\). The triangle cannot be right angled.
If two of them are even and the third one is odd, then \((side_1)^2 + (side_2)^2 = even+even=even\) and \((side_3)^2=odd\) OR \((side_1)^2 + (side_2)^2 = even+odd=odd\) and \((side_3)^2=even\). In any case \((side_1)^2 + (side_2)^2 \neq (side_3)^2\). The triangle cannot be right angled.
So, we have a definite answer to the question: the triangle is NOT right angled. Sufficient.
From above we can derive that if the lengths of all three sides of a right triangle are integers (so if a right triangle is Pythagorean Triple triangle), then the perimeter of the triangle must be even.
(2) If the triangle's area is doubled, the result is not an integer.
If the triangle were right angled, the area would be \(\frac{(leg_1)*(leg_2)}{2}\) and double that would be \(2*\frac{(leg_1)*(leg_2)}{2}=(leg_1)*(leg_2)=integer*integer=integer\). Since we are told that double the area of the triangle is NOT an integer, then the triangle is NOT right angled. Sufficient.
Answer: D.
Hope it's clear.
22 Mar 2017, 12:45
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The Three sides of triangle have length p, q and r, each an integer. Is this triangle right triangle?
Two points to be known about a Right Angled Triangle
The height and the hypotenuse of Right Angled Triangle will always be Odd Integers
The Base of the Right Angled Triangle will always be an Even Integer
Statement 1) the perimeter of the triangle is an odd integer.
Based on the above, the perimeter of a Right Angled Triangle will always be Even. Hence the Triangle in question is not a right Angled Triangle. This statement is sufficient.
Statement 2) if the triangle's area is doubled, the result is not an integer.
The Area of a Right Angled Triangle is 1/2 * b*h. On doubling this it becomes b*h. Since all sides of a Right Angles Triangle are integers, b*h will also be an integer. So in this case, the Triangle in question is not a right Angled Triangle. This statement is sufficient.
Answer is D
Two points to be known about a Right Angled Triangle
The height and the hypotenuse of Right Angled Triangle will always be Odd Integers
The Base of the Right Angled Triangle will always be an Even Integer
Statement 1) the perimeter of the triangle is an odd integer.
Based on the above, the perimeter of a Right Angled Triangle will always be Even. Hence the Triangle in question is not a right Angled Triangle. This statement is sufficient.
Statement 2) if the triangle's area is doubled, the result is not an integer.
The Area of a Right Angled Triangle is 1/2 * b*h. On doubling this it becomes b*h. Since all sides of a Right Angles Triangle are integers, b*h will also be an integer. So in this case, the Triangle in question is not a right Angled Triangle. This statement is sufficient.
Answer is D
