Question

A Pythagorean triplet whose smallest number is 8, is:
A. 8, 15, 18
B. 8, 13, 16
C. 8, 14, 17
D. 8, 15, 17

Answer 1

Hint: We use the general formula for Pythagorean triplet and equate the given number. Using the substitution method we find two other numbers that conclude the Pythagorean triplet.
* A Pythagorean triplet \[(a,b,c)\] is given by the formula \[(2m,{m^2} - 1,{m^2} + 1)\]

Complete step-by-step solution:
We take \[(a,b,c)\] as the Pythagorean triplet
We are given the smallest numbers is 8
From the three numbers \[(2m,{m^2} - 1,{m^2} + 1)\] we know \[2m\] is the smallest number
Let us assume the value of \[a = 8\]
From the formula of the Pythagorean triplet we know a triplet \[(a,b,c)\]is given by the formula\[(2m,{m^2} - 1,{m^2} + 1)\].
\[ \Rightarrow 2m = 8\]
Divide both sides of the equation by 2
\[ \Rightarrow \dfrac{{2m}}{2} = \dfrac{8}{2}\]
\[ \Rightarrow m = 4\].................… (1)
Now we know value of \[b = {m^2} - 1\]
Substitute the value of ‘m’ from equation (1) in value of ‘b’
\[ \Rightarrow b = {(4)^2} - 1\]
Square the term in RHS of the equation
\[ \Rightarrow b = 16 - 1\]
Calculate the difference in RHS of the equation
\[ \Rightarrow b = 15\]...................… (2)
Now we know value of \[c = {m^2} + 1\]
Substitute the value of ‘m’ from equation (1) in value of ‘c’
\[ \Rightarrow c = {(4)^2} + 1\]
Square the term in RHS of the equation
\[ \Rightarrow c = 16 + 1\]
Calculate the sum in RHS of the equation
\[ \Rightarrow c = 17\]............… (3)
From equations (2) and (3) the value of \[b = 15,c = 17\]
Since we know that smallest number is 8, so the Pythagorean triplet is 8, 15, 17

\[\therefore \]The correct option is D.

Note: Students might try to solve for the Pythagorean triplet by using the Pythagoras theorem and writing sum of squares of two numbers equal to square of a third number where they might take any number as 8 and try solving the equation. This is the wrong approach as we will not reach a final answer, we will just get a relation between two of the missing numbers. Since we are required to find the other two numbers we will directly use the formula for Pythagorean triplet.