Question

What are the Pythagorean triples whose one member is 20?
A. $20,97,99$
B. $20,99,101$
C. $20,197,199$
D. $20,199,201$

Answer 1

Hint: According to the question given in the question we have to determine the Pythagorean triples whose one member is 20. So, first of all we have to understand about the Pythagorean triplet which is explained as below:
Pythagorean triplet: Pythagorean triplet is consist of a three positive integers as x, y, and z such that ${x^2} + {y^2} = {z^2}$and when the sides of a triangle are Pythagorean triplet then triangle is a right angled triangle.
Now, first of all as we know that for any natural number $n > 1$and $2n,{n^2} - 1,{n^2} + 1$forms a Pythagorean triplet.
Now, we have to obtain the values of $2n,{n^2} - 1,{n^2} + 1$which are the required Pythagorean triplets.

Complete step-by-step solution:
Step 1: First of all as we know for any natural number $n > 1$ the Pythagorean triplets are:$ \Rightarrow 2n,{n^2} - 1,{n^2} + 1$……………(1)
Step 2: Now, we can obtain the first triplet by substituting the value in triplet ${n^2} - 1$. Hence,
$
   \Rightarrow {n^2} - 1 = 20 \\
   \Rightarrow {n^2} = 21.................(2)
 $
Step 3: Same as the step 2 we can obtain the first triplet by substituting the value in triplet${n^2} + 1$. Hence,
$
   \Rightarrow {n^2} + 1 = 20 \\
   \Rightarrow {n^2} = 19.................(3)
 $
Step 4: Same as the step 2 we can obtain the first triplet by substituting the value in triplet$2n$. Hence,
$
   \Rightarrow 2n = 20 \\
   \Rightarrow n = \dfrac{{20}}{2} \\
   \Rightarrow n = 10
 $
Step 5: Now, we have to substitute the value of n in the triplet as${n^2} - 1$. Hence,
$
   = {10^2} - 1 \\
   = 100 - 1 \\
   = 99
 $
Step 5: Now, same as the step 5 we have to substitute the value of n in the triplet as${n^2} + 1$. Hence,
$
   = {10^2} + 1 \\
   = 100 + 1 \\
   = 101
 $
Hence, we have obtained the Pythagorean triples whose one member is 20 are $20,99,101$.

Therefore option (B) is correct.

Note: A Pythagorean triplet always consists of all even numbers, or two odd numbers and an even number.
Pythagorean triplet is consist of a three positive integers as x, y, and z such that ${x^2} + {y^2} = {z^2}$ and when the sides of a triangle are Pythagorean triplet then triangle is a right angled triangle.