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
$$\Rightarrow {\displaystyle \frac{2m}{2}}={\displaystyle \frac{8}{2}}$$
$$\Rightarrow m=4$$.................… (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.