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In the following picture, $XY = YZ$, $\angle a = 40^\circ$, and the length opposite is $5$.

enter image description here

The problem asks to compute $XY$

We immediately dropped the dotted perpendicular and get two right triangles. The length of the bottom is 2.5 in each right triangle, and therefore $a/2 = 20^\circ$

$\sin 20 = 2.5 / XY$ or, $XY = 2.5 / \sin 20 \approx 7.3$

Their answer uses $5 / a = XY / 70$

$5 / 40 = XY / 70$ $XY = 350/40 = 8.75$

I would have thought both these answers should be the same, where did we go wrong?

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  • $\begingroup$ Their statement $5/40=XY/70$ is unjustified and wrong. $\endgroup$ – Ross Millikan Aug 3 at 19:23
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    $\begingroup$ Which book is this? I have an ACT book and I want to make sure it's not the same one? $\endgroup$ – Gnumbertester Aug 3 at 19:27
  • $\begingroup$ The book is probably going for the law of sines that $\frac 5{\sin 40} = \frac {XY}{\sin 70}$. But that the book would make such a typo and then actually calculate based on that typo does not speak well for the text. The OP was correct and the book was dead wrong. $\endgroup$ – fleablood Aug 3 at 19:27
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Get another book.

NOW!

The books answer that $\frac 5{\angle a} = \frac {XY}{\angle 70}$ is ... baseless.

There's no such similarity between triangle sides and the direct measure of angles. It's .... stupid... to think there would be.

But there is a similarity between sides and the SINES of angles.

I.e. The law of sines which would allow us to note:

that $\frac 5{\sin a} = \frac {XY}{\sin 70}$ or $\frac 5{\sin 40}=\frac {XY}{\sin 70}$ so $XY =5*\frac{\sin 70}{\sin 40} \approx 7.3$. Which ... is the same as your answer.

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  • $\begingroup$ Thanks! I thought I was losing my mind. Now I know that's true, but at least you showed us what was going on. And my son can go on to other problems. $\endgroup$ – Dov Aug 3 at 19:33
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    $\begingroup$ I might be being a little harsh on the book. But the book made a typo of $\frac 5{40} = \frac {xy}{70}$ instead of $\frac 5{\sin 40}=\frac {xy}{\sin 70}$. That's bad, but... we all make errors. But then the book made a miscalculation based on the typo. That's .... worse. I'm not sure how condemning I should be but that's the thing that should be caught. ... anyway the harm such an error can make in the utter confusion and errors it would cause and teach the students reading it is ... tragic. $\endgroup$ – fleablood Aug 3 at 19:38
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    $\begingroup$ @fleablood: Please tell us the publisher name so that we all avoid buying any books from them. $\endgroup$ – mentallurg Aug 4 at 12:34
  • $\begingroup$ DOn't ask me. Ask the OP. $\endgroup$ – fleablood Aug 4 at 15:04
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$$XY=\frac{2.5}{\sin20^{\circ}}=7.3095...$$ Another way:

$$\frac{XY}{\sin70^{\circ}}=\frac{5}{\sin40^{\circ}},$$ which gives the same result: $$XY=\frac{5\sin70^{\circ}}{\sin40^{\circ}}=7.3095...$$

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  • $\begingroup$ which is wrong according to the answer... The other way of computing it using law of signs gets us 8.75, what's going on? $\endgroup$ – Dov Aug 3 at 19:09
  • $\begingroup$ @Dov I added something. See now. $\endgroup$ – Michael Rozenberg Aug 3 at 19:12
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    $\begingroup$ @Dov: In the second approach, you've left off the sines in using the law of sines. (Which I now see that Rozenberg included in his answer as I was writing my comment.) $\endgroup$ – Dave L. Renfro Aug 3 at 19:12
  • $\begingroup$ It's the law of "sines" not "signs". So it would state that $\frac 5{\sin 40} = \frac {XY}{\sin 70}$. To misquote it, as the book did, as $\frac 5{40} = \frac {XY}{70}$ is just plain wrong as $x \ne \sin x$. $\endgroup$ – fleablood Aug 3 at 19:30
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Their answer uses 5/a=XY/70

This is wrong. There is no such rule. There is a rule of sines. Instead of 5/a=XY/70 there should be 5/sin(a)=XY/sin(70). Then you will get the same result, ~7.3.

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As others noted, the rule would be $\frac{5}{\sin a} = \frac{XY}{\sin 70°}$ and from there some editor multiplied both sides by $\sin$ and omitted the $°$ from the angle. Which is practical joke level mathematics like $\frac{16}{64}=\frac{1\not 6}{\not 64}=\frac14$. Except that in this case the result is wrong. Which means that the person responsible for doing the final substantial corrections to the text (whether its author or not) was unqualified to do so. That does not bode well for the rest of the book.

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