# Find the Pythagorean triplet whose one member is 18.

A. 28 and 20

B. 80 and 82

C. 72 and 92

D. 34 and 54

Answer

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Hint: Let us use Pythagoras theorem to solve this question. We will check each option to find the correct answer.

Complete step-by-step answer:

Now, the question is asking about the Pythagorean triplet of 18. So, we will apply Pythagoras theorem to solve the given question. Pythagoras theorem states that in a right-angled triangle, the sum of squares of base and perpendicular is equal to the square of the hypotenuse which is the largest side of a triangle while the base and perpendicular are shorter in length as compared to the hypotenuse. In simple words, if a, b, c is the length of sides of a right-angled triangle where c is the hypotenuse and a, b is the base and perpendicular respectively, then according to Pythagoras theorem,

\[{a^2} + {b^2} = {c^2}\]

Now, checking the given options by applying Pythagoras theorem in each option, taking the longest side as hypotenuse.

Checking the option (A),

Taking 28 as hypotenuse and 20 as base. Let the perpendicular be a. So, applying Pythagoras theorem,

$ \Rightarrow $ ${a^2} + {20^2} = {28^2}$

$ \Rightarrow $ ${a^2} = {28^2} - {20^2} = (28 - 20)(28 + 20)$ as, ${a^2} - {b^2} = (a - b)(a + b)$

$ \Rightarrow $ ${a^2} = (8)(48) = 384$

$ \Rightarrow $ $a = \pm \sqrt {384} = \pm 19$

${a^2} - {b^2} = (a - b)(a + b)$

$ \Rightarrow $ $a = 19$ as, side cannot be negative.

Now, we can see that 18 is given in the question as the other side but according to option (A) 19 is the other side, so option (A) is incorrect.

Checking option (B), taking 82 as hypotenuse and 80 as base. Let the perpendicular be a. So, applying Pythagoras theorem,

$ \Rightarrow $ ${a^2} + {80^2} = {82^2}$

$ \Rightarrow $ ${a^2} = {82^2} - {80^2} = (82 - 80)(82 + 80)$

$ \Rightarrow $ ${a^2} = (2)(162) = 324$

$ \Rightarrow $ $a = \pm \sqrt {324} = \pm 18$

$ \Rightarrow $ $a = 18$

Now, we can see that 18 is given in the question as the other side and according to option (B) 18 is the other side, so option (B) is correct.

Similarly, by checking option (C) and option (D) we can see that both the options are incorrect. So, the correct option is A i.e. 18 is the Pythagorean triplet of 80 and 82.

Note: It is important that you should take the largest side as hypotenuse while applying the Pythagoras theorem, if a shorter side is taken it will result in a negative term inside the under-root which is not possible. Also, to make calculation faster, always apply the formula ${a^2} - {b^2} = (a - b)(a + b)$, whenever you have to solve the question to find the Pythagorean triplet of a given number.

Complete step-by-step answer:

Now, the question is asking about the Pythagorean triplet of 18. So, we will apply Pythagoras theorem to solve the given question. Pythagoras theorem states that in a right-angled triangle, the sum of squares of base and perpendicular is equal to the square of the hypotenuse which is the largest side of a triangle while the base and perpendicular are shorter in length as compared to the hypotenuse. In simple words, if a, b, c is the length of sides of a right-angled triangle where c is the hypotenuse and a, b is the base and perpendicular respectively, then according to Pythagoras theorem,

\[{a^2} + {b^2} = {c^2}\]

Now, checking the given options by applying Pythagoras theorem in each option, taking the longest side as hypotenuse.

Checking the option (A),

Taking 28 as hypotenuse and 20 as base. Let the perpendicular be a. So, applying Pythagoras theorem,

$ \Rightarrow $ ${a^2} + {20^2} = {28^2}$

$ \Rightarrow $ ${a^2} = {28^2} - {20^2} = (28 - 20)(28 + 20)$ as, ${a^2} - {b^2} = (a - b)(a + b)$

$ \Rightarrow $ ${a^2} = (8)(48) = 384$

$ \Rightarrow $ $a = \pm \sqrt {384} = \pm 19$

${a^2} - {b^2} = (a - b)(a + b)$

$ \Rightarrow $ $a = 19$ as, side cannot be negative.

Now, we can see that 18 is given in the question as the other side but according to option (A) 19 is the other side, so option (A) is incorrect.

Checking option (B), taking 82 as hypotenuse and 80 as base. Let the perpendicular be a. So, applying Pythagoras theorem,

$ \Rightarrow $ ${a^2} + {80^2} = {82^2}$

$ \Rightarrow $ ${a^2} = {82^2} - {80^2} = (82 - 80)(82 + 80)$

$ \Rightarrow $ ${a^2} = (2)(162) = 324$

$ \Rightarrow $ $a = \pm \sqrt {324} = \pm 18$

$ \Rightarrow $ $a = 18$

Now, we can see that 18 is given in the question as the other side and according to option (B) 18 is the other side, so option (B) is correct.

Similarly, by checking option (C) and option (D) we can see that both the options are incorrect. So, the correct option is A i.e. 18 is the Pythagorean triplet of 80 and 82.

Note: It is important that you should take the largest side as hypotenuse while applying the Pythagoras theorem, if a shorter side is taken it will result in a negative term inside the under-root which is not possible. Also, to make calculation faster, always apply the formula ${a^2} - {b^2} = (a - b)(a + b)$, whenever you have to solve the question to find the Pythagorean triplet of a given number.

Last updated date: 28th May 2023

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