Help with this geometry/trigonometry common core question? I thought I could simply use the Pythagorean Theorem, but my teacher went over it today in class and explained that the correct answer is...

Help with this geometry/trigonometry common core question?

I thought I could simply use the Pythagorean Theorem, but my teacher went over it today in class and explained that the correct answer is something about the altitude and theorem. At this point, I was simply lost. Please help!

Thanks!

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steveschoen | College Teacher | (Level 1) Associate Educator

Posted on

When your teacher was talking about the theorem, it was probably the Pythagorean Theorem.  When they were talking about the altitude, they were probably talking about one of the triangles, like base, height, and hypotenuse.

As far as this problem goes, it can be very visual.  So, it may assist if you were to separate the triangles out into the three triangles.  I have that on the attachment.

There may be several ways to do this.  The thing is, you need to show that, lack of better word, using similar triangles, you can go from something like:

ad^2 + bd^2 = ab^2      to      ab^2 + bc^2 + ac^2

To do this, we could start with:

ad^2 + bd^2 = ab^2

Also, using the similar triangles, where ratios of sides are proportional.  We can write:

  ab    =   ad  
  ac          ab

Multiplying each side by "ab", we get:

ad = ab^2
         ac

Similarly, using similar triangles, we can write:

  bc    =   bd  
  ac         ab

Multiplying each side by "ab", we get:

bd = ab*bc
          ac

Substituting these into our initial Pythagoream Theorem, "ad^2 + bd^2 = ab^2", we get:

(ab^2)^2  +  (ab*bc)^2   =   ab^2
   ac^2             ac^2

ab^4   +   ab^2*bc^2   =   ab^2
ac^2              ac^2

Multiplying each side by "ac^2", we get:

ab^4  +  ab^2*bc^2   =   ab^2*ac^2

Dividing each side by "ab^2", we get:

ab^4  +  ab^2*bc^2   =   ab^2*ac^2
ab^2           ab^2                ab^2

Cancelling out ab^2 on the left side, we get:

ab^2 + bc^2 = ac^2

Which is what the question is asking you to prove.

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steveschoen's profile pic

steveschoen | College Teacher | (Level 1) Associate Educator

Posted on

Oops, sorry.  While I was doing the problem, the editor was showing the "fraction bars" between the ratios.  Everywhere where you see something like:

ab^4
ac^2

It is suppose to have a line between it, like:

ab^4

------

ac^2

They all are suppose to be ratios, aka fractions.

Sorry.

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Nolan McShea | Student, Grade 11 | (Level 2) Honors

Posted on

Thank you for the response. This is the way that I'd like to use, but however, it's not the way that our teacher told us from the answer booklet. The answer booklet does different proportions and weird, harder, longer ways to solve this problem... 

But I will still try to see if I can connect your solution to somewhere where I think it matches the answer booklet's solution. 

Thanks!

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Nolan McShea | Student, Grade 11 | (Level 2) Honors

Posted on

Oh, I think I've got it now. Thanks once again, for your help! I really dislike how this stuff is based a lot around orientation, because I'm terrible with that.

Even with Lumosity games, I remember this one game being about orientation, for matching which two shapes are identical (one of the shapes have been rotated or flipped so that it may look entirely different). It just takes me forever to get myself used to this!

Also, thanks for drawing an image to make it easier for me to understand.

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kseddy123 | College Teacher | (Level 1) Assistant Educator

Posted on

Plz check this answer this is very simple to understand hope this helps you 

plz check the attachments for the clear answer 

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