# 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

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
10,940
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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
19,523
5,399
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
3,866
596
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 Moderator

Mar 31, 2012
19,523
5,399
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 Moderator

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
930
286
NO=24*(40+10)/40=30
OP=(30^2+50^2)^0.5= 58.30952

9. ### WBahn Moderator

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

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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

Thanks everybody.

Zulfi.

11. ### dl324 Distinguished Member

Mar 30, 2015
4,593
972
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
1
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
4,593
972
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
351
1
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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972
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
351
1
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 Moderator

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.