Answer:
Second statement is true.
The lengths 7, 40 and 41 can not be sides of a right triangle. The lengths 12, 16, and 20 can be sides of a right triangle.
Step-by-step explanation:
for first part of statement
The lengths 7, 40 and 41 can not be sides of a right triangle.
If the square of long side is equal to the sum of square of other two sides
then the given length can be sides of a right triangle.
Check the given length by Pythagoras Theorem.
[tex]c^{2} =a^{2} +b^{2}[/tex]----------(1)
Let [tex]c=41[/tex] and [tex]a = 7[/tex] and [tex]b=40[/tex]
Put all the value in equation 1.
[tex]41^{2} =7^{2} +40^{2}[/tex]
[tex]1681=49+1600[/tex]
[tex]1681=1649[/tex]
Therefore, the square of long side is not equal to the sum of square of other two sides, So given lengths 7, 40 and 41 can not be sides of a right triangle.
for second part of statement.
The lengths 12, 16, and 20 can be sides of a right triangle.
Check the given length by Pythagoras Theorem.
Let [tex]c=20[/tex] and [tex]a = 12[/tex] and [tex]b=16[/tex]
[tex]20^{2} =12^{2} +16^{2}[/tex]
[tex]400=144+256[/tex]
[tex]400=400[/tex]
Therefore, the square of long side is equal to the sum of square of other two sides, So given the lengths 12, 16, and 20 can be sides of a right triangle.
Therefore, The lengths 7, 40 and 41 can not be sides of a right triangle. The lengths 12, 16, and 20 can be sides of a right triangle.
