What is the area of the right-angled triangle with hypotenuse 10 and altitude 6?

[Boris Leykin]

In a right-angled triangle hypotenuse equals 10, altitude equals 6. What is the area of the triangle?
Answers: 60, 30, 24, 16

 

[VietDao29]

Well, it really looks like a homework problem. >"<

 

[Boris Leykin]

Well, it really looks like a homework problem. >"<

Of course not, it is a standart test.

Why not 30, tell me?
Area of a triangle equals half product of altitude by base side

 

[robphy]

Of course not, it is a standart test.

Why not 30, tell me?
Area of a triangle equals half product of altitude by base side

So, what is the base?

 

[symbolipoint]

My approach may be based on a misunderstanding, so check your textbook;
but, if "altitutude" means the distance from the vertex of the right-angle to the closest point on the hypotenuse, then area would be half of base times altitude, or one half of 10 x 6, which would be 30.
As I said, I may have mishandled some of the terminology in interpreting the exercise, so check your textbook about this topic.

 

[Boris Leykin]

My approach may be based on a misunderstanding, so check your textbook;
but, if "altitutude" means the distance from the vertex of the right-angle to the closest point on the hypotenuse, then area would be half of base times altitude, or one half of 10 x 6, which would be 30.
As I said, I may have mishandled some of the terminology in interpreting the exercise, so check your textbook about this topic.

Aaaaahhhhaaa! Got you.
You are wrong, such a triangle with hypotenuse 10 and altitude to hypotenuse 6 does not exist. 24 is the correct answer. This is another http://golem.ph.utexas.edu/category/2007/06/more_mysteries_of_the_number_2.html" Oooohhhh.

 

[JohnDuck]

Aaaaahhhhaaa! Got you.
You are wrong, such a triangle with hypotenuse 10 and altitude to hypotenuse 6 does not exist. 24 is the correct answer. This is another http://golem.ph.utexas.edu/category/2007/06/more_mysteries_of_the_number_2.html" Oooohhhh.

I fail to see why this is amusing. It's only "tricky" because of ambiguous wording--if you interpret it such that one of the legs has length 6, then you'll get the answer 24 (which it seems everyone on the poll chose except you). If you interpret it as symbolipoint did, then you fall prey to a flawed premise from a trusted source. Is symbolipoint at fault because you (apparently intentionally) deceived him? Very unsporting of you, really.

 

[Boris Leykin]

I fail to see why this is amusing. It's only "tricky" because of ambiguous wording--if you interpret it such that one of the legs has length 6, then you'll get the answer 24 (which it seems everyone on the poll chose except you). If you interpret it as symbolipoint did, then you fall prey to a flawed premise from a trusted source. Is symbolipoint at fault because you (apparently intentionally) deceived him? Very unsporting of you, really.

Sorry, this was stupid idea, not amusing.

It is from V.Arnold http://ilib.mccme.ru/pdf/VIA-taskbook.htm"

6. In a right-angled triangle (in american standart exam) hypotenuse equals 10, altitude to hypotenuse equals 6. What is the area of the triangle?
For 10 years american schoolchildren successfully solved this problem, but then from Moscow russians came, no one could solve it like americans did (who gave the answer 30). Why?

 

[ObsessiveMathsFreak]

6. In a right-angled triangle (in american standart exam) hypotenuse equals 10, altitude to hypotenuse equals 6. What is the area of the triangle?

That's not the question you initially asked.

For 10 years american schoolchildren successfully solved this problem, but then from Moscow russians came, no one could solve it like americans did (who gave the answer 30). Why?

Without knowing the particulars of Russian syntax, I honestly couldn't say. However I would note that it is not stated from which part of america the students were from.

This whole question would be better asked, and be of more benefit to everyone who attempted it, if it was posed in the form of a diagram (see attachment). The question would go; In the diagram shown, the blue line has length 10, the red line has length 6. What is the area of the triangle.

Geometry becomes more "readable" when you substitute colours for words. Frankly, words whould be avoided where possible, as they can only lead to pitfalls as displayed in the Arnold question. There's a very interesting edition of euclids elements available online. It's a pity modern publishers keep churning out the same old mess with long winding paragraphs full of "the point A" "the line CD", the triangle "AFH" , etc ,etc ad nauseum. Colour really helps clarify things.

 

[AlephZero]

Boris, your proposed solution with area 30 does not exist.

Draw the circumscribing circle of the triangle.
The hypotenuse of the triangle = the diameter of the circle = 10.
The maximum altitude (measured perpendicular to the hypotenuse) is the radius of the circle = 5.
Your "altitude = 6" is impossible.
The maximum possible area for a right triangle with hypotenuse 10 is 10*5/2 = 25.

I read the original question as meaning the altitude (vertical side) = 6, the hypotenuse (sloping) = 10,
By Pythagoras theorem the base (horizontal side) = 8.
Area = 6*8/2 = 24.