In the diagram below, P is a point inside square ABCD.  Find x.


 May 4, 2023

Since AC is the diagonal of square ABCD we know that it is a right triangle.

There are 2 methods.


1st Method(45-45-90 Triangle)


Since AC is a diagonal of square ABCD. We know that triangle ABC is a 45-45-90 triangle.
Using our knowledge that it is a 45-45-90 triangle and the ratios will go 1:1:\(x = \sqrt{2}\). We have x:x:2 = 1:1:\(\sqrt{2}\) 
Dividing 2 by \(\sqrt{2}\) we get \(2/\sqrt{2}\) which rationalizing the denominator equals \(x = \sqrt{2}\).

\(x = \sqrt{2}\)

Also rationalizing the demoninator involves multiplying a fraction with an irrational denominator with itself/itself. For example rationalizing \(2/\sqrt{2}\) in this case would involve multiplying the fraction by \(\sqrt{2}/\sqrt{2} = 1\)


Another method is easier for beginners(Pythagorean Theorem)


Knowing that the figure above is a square we know BC is also equal to x. Using the pythagorean theorem we get \(x^2 + x^2 = 2^2 or 2x^2 = 4\) . Dividing by 2 on both sides we get \(x^2 = 2\). Squarerooting on both sides we get \(x = +- \sqrt{2}\) However, since a length cannot have negative length.


\(x = \sqrt{2}\)

 May 4, 2023

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