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1

$begingroup$

In the following picture, XY = YZ. The angle a = 40 degrees, and the length opposite is 5.

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?

geometry trigonometry fake-proofs

share|cite|improve this question

edited 1 hour ago

N. Bar

60911 silver badge1717 bronze badges

asked 8 hours ago

DovDov

17288 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Get another book. NOW! The books answer that $frac 5angle a = frac XY70$ is ... baseless. It seems like they want to use the law of sins that says $frac 5sin a = frac xysin 70$ or $frac 5sin 40=frac XYsin 70$ so $XY =5*fracsin 70sin 40 approx 7.3$. Which ... is the same as yours.
  $endgroup$
  – fleablood
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Their statement $5/40=XY/70$ is unjustified and wrong.
  $endgroup$
  – Ross Millikan
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Which book is this? I have an ACT book and I want to make sure it's not the same one?
  $endgroup$
  – Gnumbertester
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The book is probably going for the law of sines that $frac 5sin 40 = frac XYsin 70$. 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
  7 hours ago
  

  
  
  
  

add a comment | 

1

$begingroup$

In the following picture, XY = YZ. The angle a = 40 degrees, and the length opposite is 5.

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?

geometry trigonometry fake-proofs

share|cite|improve this question

edited 1 hour ago

N. Bar

60911 silver badge1717 bronze badges

asked 8 hours ago

DovDov

17288 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Get another book. NOW! The books answer that $frac 5angle a = frac XY70$ is ... baseless. It seems like they want to use the law of sins that says $frac 5sin a = frac xysin 70$ or $frac 5sin 40=frac XYsin 70$ so $XY =5*fracsin 70sin 40 approx 7.3$. Which ... is the same as yours.
  $endgroup$
  – fleablood
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Their statement $5/40=XY/70$ is unjustified and wrong.
  $endgroup$
  – Ross Millikan
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Which book is this? I have an ACT book and I want to make sure it's not the same one?
  $endgroup$
  – Gnumbertester
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The book is probably going for the law of sines that $frac 5sin 40 = frac XYsin 70$. 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
  7 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

In the following picture, XY = YZ. The angle a = 40 degrees, and the length opposite is 5.

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?

geometry trigonometry fake-proofs

share|cite|improve this question

edited 1 hour ago

N. Bar

60911 silver badge1717 bronze badges

asked 8 hours ago

DovDov

17288 bronze badges

$endgroup$

share|cite|improve this question

edited 1 hour ago

N. Bar

60911 silver badge1717 bronze badges

asked 8 hours ago

DovDov

17288 bronze badges

share|cite|improve this question

edited 1 hour ago

N. Bar

60911 silver badge1717 bronze badges

asked 8 hours ago

DovDov

17288 bronze badges

edited 1 hour ago

N. Bar

60911 silver badge1717 bronze badges

edited 1 hour ago

N. Bar

60911 silver badge1717 bronze badges

asked 8 hours ago

DovDov

17288 bronze badges

asked 8 hours ago

DovDov

17288 bronze badges

2 Answers
2

active

oldest

votes

4

$begingroup$

Get another book.

NOW!

The books answer that $frac 5angle a = frac XYangle 70$ 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 $frac 5sin a = frac XYsin 70$ or $frac 5sin 40=frac XYsin 70$ so $XY =5*fracsin 70sin 40 approx 7.3$. Which ... is the same as your answer.

share|cite|improve this answer

answered 7 hours ago

fleabloodfleablood

77.5k22 gold badges2828 silver badges9595 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I might be being a little harsh on the book. But the book made a typo of $frac 540 = frac xy70$ instead of $frac 5sin 40=frac xysin 70$. 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
  7 hours ago
  
  

  
  
  
  

add a comment | 

3

$begingroup$

$$XY=frac2.5sin20^circ=7.3095...$$
Another way:

$$fracXYsin70^circ=frac5sin40^circ,$$ which gives the same result:
$$XY=frac5sin70^circsin40^circ=7.3095...$$

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

Michael RozenbergMichael Rozenberg

122k2020 gold badges105105 silver badges210210 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  8 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Dov I added something. See now.
  $endgroup$
  – Michael Rozenberg
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @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
  8 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It's the law of "sines" not "signs". So it would state that $frac 5sin 40 = frac XYsin 70$. To misquote it, as the book did, as $frac 540 = frac XY70$ is just plain wrong as $x ne sin x$.
  $endgroup$
  – fleablood
  7 hours ago
  

  
  
  
  

add a comment | 

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4

$begingroup$

Get another book.

NOW!

The books answer that $frac 5angle a = frac XYangle 70$ 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 $frac 5sin a = frac XYsin 70$ or $frac 5sin 40=frac XYsin 70$ so $XY =5*fracsin 70sin 40 approx 7.3$. Which ... is the same as your answer.

share|cite|improve this answer

answered 7 hours ago

fleabloodfleablood

77.5k22 gold badges2828 silver badges9595 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I might be being a little harsh on the book. But the book made a typo of $frac 540 = frac xy70$ instead of $frac 5sin 40=frac xysin 70$. 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
  7 hours ago
  
  

  
  
  
  

add a comment | 

4

$begingroup$

Get another book.

NOW!

The books answer that $frac 5angle a = frac XYangle 70$ 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 $frac 5sin a = frac XYsin 70$ or $frac 5sin 40=frac XYsin 70$ so $XY =5*fracsin 70sin 40 approx 7.3$. Which ... is the same as your answer.

share|cite|improve this answer

answered 7 hours ago

fleabloodfleablood

77.5k22 gold badges2828 silver badges9595 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  7 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I might be being a little harsh on the book. But the book made a typo of $frac 540 = frac xy70$ instead of $frac 5sin 40=frac xysin 70$. 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
  7 hours ago
  
  

  
  
  
  

add a comment | 

$begingroup$

Get another book.

NOW!

The books answer that $frac 5angle a = frac XYangle 70$ 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 $frac 5sin a = frac XYsin 70$ or $frac 5sin 40=frac XYsin 70$ so $XY =5*fracsin 70sin 40 approx 7.3$. Which ... is the same as your answer.

share|cite|improve this answer

answered 7 hours ago

fleabloodfleablood

77.5k22 gold badges2828 silver badges9595 bronze badges

$endgroup$

share|cite|improve this answer

answered 7 hours ago

fleabloodfleablood

77.5k22 gold badges2828 silver badges9595 bronze badges

answered 7 hours ago

fleabloodfleablood

77.5k22 gold badges2828 silver badges9595 bronze badges

answered 7 hours ago

fleabloodfleablood

77.5k22 gold badges2828 silver badges9595 bronze badges

3

$begingroup$

$$XY=frac2.5sin20^circ=7.3095...$$
Another way:

$$fracXYsin70^circ=frac5sin40^circ,$$ which gives the same result:
$$XY=frac5sin70^circsin40^circ=7.3095...$$

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

Michael RozenbergMichael Rozenberg

122k2020 gold badges105105 silver badges210210 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  8 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Dov I added something. See now.
  $endgroup$
  – Michael Rozenberg
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @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
  8 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It's the law of "sines" not "signs". So it would state that $frac 5sin 40 = frac XYsin 70$. To misquote it, as the book did, as $frac 540 = frac XY70$ is just plain wrong as $x ne sin x$.
  $endgroup$
  – fleablood
  7 hours ago
  

  
  
  
  

add a comment | 

3

$begingroup$

$$XY=frac2.5sin20^circ=7.3095...$$
Another way:

$$fracXYsin70^circ=frac5sin40^circ,$$ which gives the same result:
$$XY=frac5sin70^circsin40^circ=7.3095...$$

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

Michael RozenbergMichael Rozenberg

122k2020 gold badges105105 silver badges210210 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  8 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Dov I added something. See now.
  $endgroup$
  – Michael Rozenberg
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @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
  8 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It's the law of "sines" not "signs". So it would state that $frac 5sin 40 = frac XYsin 70$. To misquote it, as the book did, as $frac 540 = frac XY70$ is just plain wrong as $x ne sin x$.
  $endgroup$
  – fleablood
  7 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

$$XY=frac2.5sin20^circ=7.3095...$$
Another way:

$$fracXYsin70^circ=frac5sin40^circ,$$ which gives the same result:
$$XY=frac5sin70^circsin40^circ=7.3095...$$

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

Michael RozenbergMichael Rozenberg

122k2020 gold badges105105 silver badges210210 bronze badges

$endgroup$

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

Michael RozenbergMichael Rozenberg

122k2020 gold badges105105 silver badges210210 bronze badges

answered 8 hours ago

Michael RozenbergMichael Rozenberg

122k2020 gold badges105105 silver badges210210 bronze badges

answered 8 hours ago

Michael RozenbergMichael Rozenberg

122k2020 gold badges105105 silver badges210210 bronze badges