Question

Which of the following is a Pythagorean Triplet, whose one of the members is 34?
(a) 34, 278, 290
(b) 34, 288, 291
(c) 34, 288, 290
(d) 35, 288, 290

Answer 1

Hint: Check each of the given options if they satisfy the conditions for a Pythagorean Triplet, which says that any three positive integers a, b and c form a Pythagorean Triplet if the sum of square of two numbers is equal to the square of the third number, i.e., \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\].
Complete step-by-step solution -
We have to find a Pythagorean Triplet whose one of the members is 34. We will check each of the options if they form a Pythagorean Triplet or not.
We know that any three positive integers a, b and c form a Pythagorean Triplet if the sum of square of two numbers is equal to the square of the third number, i.e., \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\].
We will now check the first option, which has the numbers 34, 278 and 290.
We have \[{{\left( 34 \right)}^{2}}=1156\], \[{{\left( 278 \right)}^{2}}=77284\] and \[{{\left( 290 \right)}^{2}}=84100\]. Adding \[{{\left( 34 \right)}^{2}}\] and \[{{\left( 278 \right)}^{2}}\], we have \[{{\left( 34 \right)}^{2}}+{{\left( 278 \right)}^{2}}=1156+77284=78440\]. We observe that \[{{\left( 34 \right)}^{2}}+{{\left( 278 \right)}^{2}}=78440\ne 84100={{\left( 290 \right)}^{2}}\].
Thus, 34, 278 and 290 don’t form a Pythagorean Triplet. Hence, option (a) is incorrect.
We will now check the second option, which has the numbers 34, 288 and 291.
We have \[{{\left( 34 \right)}^{2}}=1156\], \[{{\left( 288 \right)}^{2}}=82944\] and \[{{\left( 291 \right)}^{2}}=84681\]. Adding \[{{\left( 34 \right)}^{2}}\] and \[{{\left( 288 \right)}^{2}}\], we have \[{{\left( 34 \right)}^{2}}+{{\left( 288 \right)}^{2}}=1156+82944=84100\]. We observe that \[{{\left( 34 \right)}^{2}}+{{\left( 288 \right)}^{2}}=84100\ne 84681={{\left( 291 \right)}^{2}}\].
Thus, 34, 288 and 291 don’t form a Pythagorean Triplet. Hence, option (b) is incorrect.
We will now check the third option, which has the numbers 34, 288 and 290.
We have \[{{\left( 34 \right)}^{2}}=1156\], \[{{\left( 288 \right)}^{2}}=82944\] and \[{{\left( 290 \right)}^{2}}=84100\]. Adding \[{{\left( 34 \right)}^{2}}\] and \[{{\left( 288 \right)}^{2}}\], we have \[{{\left( 34 \right)}^{2}}+{{\left( 288 \right)}^{2}}=1156+82944=84100\]. We observe that \[{{\left( 34 \right)}^{2}}+{{\left( 288 \right)}^{2}}=84100=84100={{\left( 290 \right)}^{2}}\].
Thus, 34, 288 and 291 form a Pythagorean Triplet. Hence, option (c) is correct.
We will now check the fourth option, which has the numbers 35, 288 and 290. We observe that 34 is not a part of this Pythagorean Triplet and we are finding a Pythagorean Triplet for 34.
Thus, option (d) is incorrect.
Hence, the Pythagorean Triplet containing 34 is 34, 288 and 290, which is option (c).

Note: One must clearly know the definition of Pythagorean Triplet. The members of Pythagorean Triplet represent the lengths of sides of the right angles triangle, with the greatest value being equal to Hypotenuse. It’s necessary to check if each of the options represents a Pythagorean Triplet or not.