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.

 

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°

 

They are similar triangles.

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

 

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.

 

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.

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).

 

These days a 9-year old is a valuable technology resource; they know and remember things we have forgotten.

 

These days a 9-year old is a valuable technology resource; they know and remember things we have forgotten.

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.

 

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

 

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.

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

 

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.

 

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.)

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.

 

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

Zulfi.

 

I used the pythagorous first and then the ratio. Is there any prob with my approach?

Not really. Except that I could determine ON in my head. Would have needed a calculator to do the square root though...

 

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

Zulfi.

 

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

Zulfi.

Are you allowed to bring your 9-years old sister?

 

In exam i wont be having calculator.

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.

 

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.

 

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.

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