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Problem Hello mathematicians,

I was given this question by my teacher and after spending a couple of hours looking over it have not been able to solve it.

I understand it involves radians which I have attempted to learn through a few quick online courses, which have only helped me in understanding:

$\frac{3*\pi}4 = 135°$

Which is not helpful in solving the question.

The annoying thing is this is in a chapter titled Pythagoras Theorem and it Converse. Yet I am having a lot of trouble figuring it out! As always the textbook does not include answers as it is a show question.

Any pointers are appreciated.

Monacraft
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2 Answers2

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Here is something to get you started:

Place a point $Q$ on the segment $AB$ such that $AB$ and $PQ$ meet at right angles:

enter image description here

Let us introduce the variables $s=|AD|,x=|AQ|,y=|PQ|$. Then $s$ is the side length of the square. We then must have $$ \begin{align} x^2+y^2&=4\\ (s-y)^2+x^2&=1\\ (s-x)^2+y^2&=9 \end{align} $$ From the first equation we get $x^2=4-y^2$, and substituting this into the second equation we can solve for $y$ to have $y=\frac{s^2+3}{2s}$. Similarly, we may substitute $y^2=4-x^2$ into the third equation and solve for $x$ to have $x=\frac{s^2-5}{2s}$. With this, the first equation becomes $$ \left(\frac{s^2-5}{2s}\right)^2+\left(\frac{s^2+3}{2s}\right)^2=4 $$ which after multiplying both sides by $4s^2$ and using the substitution $t=s^2$ yields the quadratic equation $$ (t-5)^2+(t+3)^2=16t\iff s^2=t=5\pm2\sqrt 2 $$ Now, since $x=\frac{s^2-5}{2s}>0$ it must be $s^2=5+2\sqrt 2$. Therefore we get $$ \begin{align} \cos\theta&=\frac{1^2+2^2-s^2}{2\cdot 1\cdot 2}\\ &=-\frac{\sqrt 2}2 \end{align} $$ which can easily be derived or found in a table to be equivalent to $\theta=\frac{3\pi}4$.

String
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So turns out it is possible to do this question with Pythagoras theorem alone!

I found the proof online here

Diagram

However I do not imagine my self ever coming up with this in an exam.

Regardless thank you String and Archaick for helping me out.

Monacraft
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