Do the side lengths 7, 24, and 25 form a right triangle? Why or why not?
2 Answers
Yes, as sum of squares of smaller two sides is equal to square on the largest side, i.e.
Yes, which we can deduce from Pythagoras theorem...
Explanation:
Note on terms
In North America a triangle containing a right angle is called a "right triangle".
Elsewhere, it is called a "right-angled triangle".
I will use the term "right-angled triangle", but please read "right triangle" if you prefer.
Pythagoras' theorem
Consider the following diagram:
The area of the large outer square is equal to the area of the small, tilted square plus the area of the four right-angled triangles...
#(a+b)^2 = c^2+4*(ab)/2#
Multiplying this out, we get:
#a^2+2ab+b^2 = c^2+2ab#
Then subtracting
#a^2+b^2 = c^2#
Note that:
-
In order for the diagram to apply we only required that
#a# ,#b# ,#c# be the lengths of the sides of a right-angled triangle. Therefore any such right-angled triangle will satisfy#a^2+b^2=c^2# . -
Conversely, if
#a# ,#b# and#c# are positive numbers satisfying#a^2+b^2=c^2# , then we can construct such a diagram and observe the right-angled triangles.
Therefore three positive numbers
Example
#7^2+24^2 = 49+576 = 625 = 25^2#
Hence we can deduce that a triangle with sides