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# help plz

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

May 4, 2023

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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