Do the side lengths 7, 24, and 25 form a right triangle? Why or why not?

2 Answers
Mar 23, 2017

Yes, as sum of squares of smaller two sides is equal to square on the largest side, i.e. #7^2+24^2=25^2#. This comes from Pythagoras theorem.

Mar 23, 2017

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.

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Pythagoras' theorem

Consider the following diagram:

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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 #2ab# from both sides, we find:

#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 #a#, #b# and #c# are the lengths of the sides of a right-angled triangle with #c# the hypotenuse if and only if #a^2+b^2 = c^2#.

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Example

#7^2+24^2 = 49+576 = 625 = 25^2#

Hence we can deduce that a triangle with sides #7, 24, 25# is a right-angled triangle.