write a Pythagorean triplet if one number is 8
Answers
Step-by-step explanation:
It is given that smallest number is 8. ... Now, second number = m² -1 = 4² - 1 = 15. and, third number = m² + 1 = 4² + 1 = 17. Hence, the triplet is 8, 15 and 17
Answer:
The Pythagoras theorem states that, "the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides".
The basic formula to calculate hypotenuse is a² + b² = c², where c represents the length of the hypotenuse and a and b are the lengths of the triangle's other two sides.
If the length of both a and b are known, then c can be calculated as
In this triangle, c2=a2 + b2
Although Pythagoras theorem is applicable in case of a right-angled triangle only, yet it has a lot of direct and indirect applications.
Classification of triangles on Pythagoras theorem:
If a2 + b2 < c2, then the triangle would be an obtuse-angled triangle.
If a2 + b2 > c2, then the triangle would be an acute-angled triangle.
Let us see its application in solving questions.
Example: Find the number of acute triangles that can be formed with two of its sides equal to 8, 15.
Solution: For an acute triangle, we have condition a2 + b2 > c2
Other condition that we'll use is: The sum of 2 sides in a triangle is greater than the third side.
Let the third side be x.
Applying the first condition, we get
82+x2 > 152; 82+152 >x2; x2+15>82
x2>225-64 ⇒ x2 >161 => Since x is an integer, x=13,14,15,16,17,....
82+152 >x2 ⇒ 289>x2 => x=0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16.
x2+15>82⇒x>0
Applying the second condition, we get
8+15>x; 8+x>15; 15+x>8
⇒x<23; x>7; x> -7
&rArr Combining the two conditions, we get x=13, 14, 15, 16.
Hence we get 4 cases. Answer= 4
How to Form a Pythagorean Triplet
If the number is odd: Square the number N and then divide it by 2. Take the integer that is immediately before and after that number i.e. (N2/2 -0.5) and (N2/2 +0.5).
Pythagorean triplet= N, (N2/2 -0.5), (N2/2 +0.5)
Example: Take number N=3.
On squaring the number, we get 9.
Now take the half of it. 9/2= 4.5
The integer immediately before 4.5= 4; the integer immediately after 4.5= 5
Therefore, the triplet is 3, 4, 5.
If the number is even: Take the half of that number N and then square it.
Pythagorean triplet= N, (N/2)2-1, (N/2)2+1
Example: Take number N=8
Half of N=4.
Pythagorean triplet= 8, (42-1), (42+1) i.e. 8, 15, 17
Step-by-step explanation:
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