Find lengths of sides of inner triangle

Discussion in 'Homework Help' started by zulfi100, Jul 12, 2017.

  1. zulfi100

    Thread Starter Active Member

    Jun 7, 2012
    351
    1
    Hi,
    I have a question from a book which says:

    What are the lengths of sides NO and OP in triangle NOP?

    See the attached figure

    I don’t think that the inner triangle is the 30-60-90 triangle so we cant use the eq of 30-60-90 triangle. However we can use the pythagorous formula to determine the length of hypotnuse of inner triangle. Let X be the other end of hypotenuse of inner triangle:


    (40) * (40) + (24) * (24) = XP * XP

    So XP = 46.64
    ets length of sides of outer triangle p260Q8.jpg
    Sorry I cant figure out how to attempt this question.


    Somebody please guide me how to solve it.

    Zulfi.


     
  2. Papabravo

    Expert

    Feb 24, 2006
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    The angles are right angles and the triangles are similar, so:
    Use 24/40 = tan(∠NOP). Then side NO is 50*(24/40) = 30. Side OP is then SQRT(30^2 + 50^2)= 58.31

    ∠NOP ≈ 30.96°
     
    Last edited: Jul 12, 2017
  3. WBahn

    Moderator

    Mar 31, 2012
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    They are similar triangles.

    If you don't know the implications of two triangles being similar -- then look it up.
     
  4. shteii01

    AAC Fanatic!

    Feb 19, 2010
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    Both triangles share angle NPO. Find what it is.

    Now your large triangle has two known angles and one known side. That is enough to find the rest.
     
  5. WBahn

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    Mar 31, 2012
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    No need for any trig at all (which is good since problems like this are usually encountered well before some gets to trig).

    The length of ON is a simple mental calculation that my nine-year old daughter was able to do, albeit with a bit of coaching on how to think about using the fact that the triangles are similar. Finding the hypotenuse of either is currently beyond her, but after I gave a simplified version of what the Pythagorean Theorem requires, she was able to quickly figure out which two integer lengths the hypotenuses had to fall between. She even came up with a very crude approach to doing a binary search for the answer on her own (though that was admittedly a bit painful to watch).
     
  6. Papabravo

    Expert

    Feb 24, 2006
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    These days a 9-year old is a valuable technology resource; they know and remember things we have forgotten.
     
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  7. MrChips

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    Oct 2, 2009
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    Also a 9-year old hasn't been programmed like a 60-year old and can provide some refreshing, enlightening, and entertaining perspectives in life.
     
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  8. Bordodynov

    Active Member

    May 20, 2015
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    NO=24*(40+10)/40=30
    OP=(30^2+50^2)^0.5= 58.30952
     
  9. WBahn

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    Mar 31, 2012
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    With particular emphasis on the "entertaining", more days than not. :D
     
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  10. zulfi100

    Thread Starter Active Member

    Jun 7, 2012
    351
    1
    Hi,
    Thanks everybody. Thanks PapaBravo for his solution. However, i checked the book and found that it can be solved by relating the sides of the triangles because the triangles are similar.
    First i found XP. Note X is a point on the hypotenuse OP intersected by the altitude of smaller triangle and it was 46.64 as i showed earlier. Then i started reading the theory from the book and i was able to make a following relationship: (Note AP is the base of smaller triangle, AX is the altitude and XP is the hypotenuse of smaller triangle.) So

    OP/XP = NP/AP

    OP/46.64 = 50/40

    OP = 58.3.

    Now

    (58.3)^2 = (50) ^2 + (NO)^2

    (NO)^2 = 898

    NO = 29.98


    Answers are correct.

    Thanks everybody.


    Zulfi.
     
  11. dl324

    Distinguished Member

    Mar 30, 2015
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    I would have used the ratio of the two base segments to determine ON. Then use the Pythagorean theorem to find OP.

    As @Bordodynov did in post #8 and @WBahn probably did with his 9 year old.
     
  12. zulfi100

    Thread Starter Active Member

    Jun 7, 2012
    351
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    Hi,
    I used the pythagorous first and then the ratio. Is there any prob with my approach?

    Zulfi.
     
  13. dl324

    Distinguished Member

    Mar 30, 2015
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    Not really. Except that I could determine ON in my head. Would have needed a calculator to do the square root though...
     
  14. zulfi100

    Thread Starter Active Member

    Jun 7, 2012
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    Hi,
    Thanks. Good idea. In exam i wont be having calculator.

    Zulfi.
     
  15. atferrari

    AAC Fanatic!

    Jan 6, 2004
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    Are you allowed to bring your 9-years old sister?
     
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  16. dl324

    Distinguished Member

    Mar 30, 2015
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    From some aspects, that could be better. If you did the ratio in your head to find OP, you would have come up with 30; not 29.98.
     
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  17. zulfi100

    Thread Starter Active Member

    Jun 7, 2012
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    Hi,
    <Are you allowed to bring your 9-years old sister?>

    Right now i dont have. But I think 9 years is a good age of maths. I used to get 100/100 when i was 9.

    Zulfi.
     
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  18. WBahn

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    Mar 31, 2012
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    You might try rereading Posts #3 and #5. Even Post #2 points out that they are similar triangles.
     
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