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I make CAD drawings of buildings.

Sometimes I need to draw a bay window and I end up with a geometry problem I can't solve easily and it slows down my work.

Here is a poor quality sketch of one of these problems:

bay window plan view

The whole window takes up $2125$ mm of wall length. Each of the sloped panes has half the length of the front pane. I need to draw the window from the front.

I know that $2a + 2b = 2125$, that $b^2 + 40^2 = a^2$, and even that for some angle, $\theta$, $$\sin\theta = \frac{b}{a}$$ $$\cos\theta = \frac{40}{a}$$ and $$\tan\theta = \frac{b}{40}$$ but I am not much good at algebra and I am not able to solve any of these equations :(

How should I approach this type of problem? Can it be solved or do I need more measurements?

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    $\begingroup$ Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. $\endgroup$ – José Carlos Santos Jun 18 '19 at 10:06
  • $\begingroup$ @JoséCarlosSantos thanks a million! It was well worth asking the question just to learn that ^_^ $\endgroup$ – Zanna Jun 18 '19 at 10:13
  • $\begingroup$ It doesn't cornern math, but is that 40 supposed to be 40mm or 40cm? If it's 40mm, then this bay window is very shallow, compared to how wide it is. $\endgroup$ – Adam Latosiński Jun 18 '19 at 10:26
  • $\begingroup$ @AdamLatosiński it's 40mm. Yes, it's kind of a fake bay window I guess. But in my experience there is no standard for this angle - I draw many bay windows and they come in all shapes. $\endgroup$ – Zanna Jun 18 '19 at 10:33
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If you have two unknowns and two linearly independent equations, you can find the unknowns. Here, two equations are

$$2a+2b = 2125\ \ \ \ \ \ (1)$$ $$b^2+1600=a^2\ \ \ \ \ \ \ (2)$$

Now, what you should know is that we have $a^2-b^2 = (a+b)(a-b)$. From $(1)$, we have $a+b = 1062.5$. And $(2)$ gives $a^2-b^2 = 1600 \implies (a+b)(a-b) = 1600$. From here, you can find $a-b$ and then solving it by knowing $a+b$ is easy.

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  • $\begingroup$ Thanks so much for this insight. It will help me a lot in the future $\endgroup$ – Zanna Jun 18 '19 at 10:31
  • $\begingroup$ You're welcome :) Good luck! $\endgroup$ – ArsenBerk Jun 18 '19 at 10:32
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As you've noticed you have $$b = \sqrt{a^2-40^2} $$ so $$ 2125 = 2a + 2 \sqrt{a^2-40^2} $$ to solve such equation, you move $2a$ to the left side and take a square of both sides: $$ (2125-2a)^2 = 4(a^2-40^2)$$ $$ 2125^2 - 4\cdot 2125 a + 4a^2 = 4a^2- 4\cdot 40^2$$ $$ -4\cdot 2125 a = -2125^2 - 4\cdot 40^2$$ $$ a = \frac{2125^2 + 4\cdot 40^2}{4\cdot 2125} $$ from this you can find the rest of the dimensions easily.

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  • $\begingroup$ This is what I was trying to do, but I kept going wrong. Thanks a lot for the demonstration. $\endgroup$ – Zanna Jun 18 '19 at 10:24

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