Answer
Verified
382.5k+ views
Hint: To understand the term ‘Pythagorean triplet’ we will consider a right angle triangle. Now, we will take some examples of the cases where we will assign some positive integers as the lengths of the base and the perpendicular and use the Pythagoras theorem to find the length of the hypotenuse. If they turn out to be an integer then the triplet will be known as the Pythagorean triplet otherwise not.
Complete step by step solution:
Here we have been asked about the term ‘Pythagorean Triplet’. To understand this term we need to consider a right angle triangle, so let us draw a right angle triangle.
Now, here we can say that AB is the perpendicular (P), BC is the base (B) and AC is the hypotenuse (H) as per the convention. So, a Pythagorean Triplet is defined as the set of three positive integers that satisfies the Pythagoras theorem given as ${{H}^{2}}={{P}^{2}}+{{B}^{2}}$. In case any of the sides is not an integer even after satisfying the theorem then that triplet is not called the Pythagorean Triplet. Let us take a few examples.
(i) Consider the length of the base BC as 3 cm and the length of the perpendicular AB as 4 cm, so using the Pythagoras theorem we get,
$\begin{align}
& \Rightarrow A{{C}^{2}}={{4}^{2}}+{{3}^{2}} \\
& \Rightarrow A{{C}^{2}}=25 \\
& \Rightarrow AC=5cm \\
\end{align}$
Clearly 3 cm, 4 cm and 5 cm is a Pythagorean Triplet.
(ii) Consider the length of the base BC as 6 cm and the length of the perpendicular AB as 8 cm, so using the Pythagoras theorem we get,
$\begin{align}
& \Rightarrow A{{C}^{2}}={{8}^{2}}+{{6}^{2}} \\
& \Rightarrow A{{C}^{2}}=100 \\
& \Rightarrow AC=10cm \\
\end{align}$
Clearly 6 cm, 8 cm and 10 cm is a Pythagorean Triplet.
(iii) Consider the length of the base BC as 12 cm and the length of the perpendicular AB as 5 cm, so using the Pythagoras theorem we get,
$\begin{align}
& \Rightarrow A{{C}^{2}}={{5}^{2}}+{{12}^{2}} \\
& \Rightarrow A{{C}^{2}}=169 \\
& \Rightarrow AC=13cm \\
\end{align}$
Clearly 5 cm, 12 cm and 13 cm is a Pythagorean Triplet.
(iv) Consider the length of the base BC as 10 cm and the length of the perpendicular AB as 5 cm, so using the Pythagoras theorem we get,
$\begin{align}
& \Rightarrow A{{C}^{2}}={{10}^{2}}+{{5}^{2}} \\
& \Rightarrow A{{C}^{2}}=125 \\
& \Rightarrow AC=5\sqrt{5}cm \\
\end{align}$
Since $5\sqrt{5}$ is not an integer so we can say that 5 cm, 10 cm and $5\sqrt{5}cm$ is not a Pythagorean Triplet.
Note: Remember the basic terms used in the above solution. Note that there are infinite Pythagorean triplets so you can take many examples. Even if we know the length of one side of a right triangle and one of the angles other than the 90 degrees then we can determine the measure of all the sides using trigonometry.
Complete step by step solution:
Here we have been asked about the term ‘Pythagorean Triplet’. To understand this term we need to consider a right angle triangle, so let us draw a right angle triangle.
Now, here we can say that AB is the perpendicular (P), BC is the base (B) and AC is the hypotenuse (H) as per the convention. So, a Pythagorean Triplet is defined as the set of three positive integers that satisfies the Pythagoras theorem given as ${{H}^{2}}={{P}^{2}}+{{B}^{2}}$. In case any of the sides is not an integer even after satisfying the theorem then that triplet is not called the Pythagorean Triplet. Let us take a few examples.
(i) Consider the length of the base BC as 3 cm and the length of the perpendicular AB as 4 cm, so using the Pythagoras theorem we get,
$\begin{align}
& \Rightarrow A{{C}^{2}}={{4}^{2}}+{{3}^{2}} \\
& \Rightarrow A{{C}^{2}}=25 \\
& \Rightarrow AC=5cm \\
\end{align}$
Clearly 3 cm, 4 cm and 5 cm is a Pythagorean Triplet.
(ii) Consider the length of the base BC as 6 cm and the length of the perpendicular AB as 8 cm, so using the Pythagoras theorem we get,
$\begin{align}
& \Rightarrow A{{C}^{2}}={{8}^{2}}+{{6}^{2}} \\
& \Rightarrow A{{C}^{2}}=100 \\
& \Rightarrow AC=10cm \\
\end{align}$
Clearly 6 cm, 8 cm and 10 cm is a Pythagorean Triplet.
(iii) Consider the length of the base BC as 12 cm and the length of the perpendicular AB as 5 cm, so using the Pythagoras theorem we get,
$\begin{align}
& \Rightarrow A{{C}^{2}}={{5}^{2}}+{{12}^{2}} \\
& \Rightarrow A{{C}^{2}}=169 \\
& \Rightarrow AC=13cm \\
\end{align}$
Clearly 5 cm, 12 cm and 13 cm is a Pythagorean Triplet.
(iv) Consider the length of the base BC as 10 cm and the length of the perpendicular AB as 5 cm, so using the Pythagoras theorem we get,
$\begin{align}
& \Rightarrow A{{C}^{2}}={{10}^{2}}+{{5}^{2}} \\
& \Rightarrow A{{C}^{2}}=125 \\
& \Rightarrow AC=5\sqrt{5}cm \\
\end{align}$
Since $5\sqrt{5}$ is not an integer so we can say that 5 cm, 10 cm and $5\sqrt{5}cm$ is not a Pythagorean Triplet.
Note: Remember the basic terms used in the above solution. Note that there are infinite Pythagorean triplets so you can take many examples. Even if we know the length of one side of a right triangle and one of the angles other than the 90 degrees then we can determine the measure of all the sides using trigonometry.
Recently Updated Pages
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Give 10 examples for herbs , shrubs , climbers , creepers
Why is there a time difference of about 5 hours between class 10 social science CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE