Puzzle time - the Triangle
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wrote on 28 Dec 2020, 04:42 last edited by jon-nyc
A teacher (lets call him Klaus) used to give this problem as part of his elementary geometry class:
A right triangle has hypotenuse (BC) of length 10, and a height (AP) of length 6. What is the area?
His average students (lets call them Horace and Ax) got full credit for the answer 1/2 * 10 * 6 = 30.
But a really smart kid (lets call her Taiwan girl) couldn’t solve the problem.
Why not?
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wrote on 28 Dec 2020, 06:00 last edited by Klaus
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There are three hypothenuses. Which one has the length 10?
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wrote on 28 Dec 2020, 10:01 last edited by
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Is it something as boring as "it isn't specified that AP is the height, so it could be something slightly different"?
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wrote on 28 Dec 2020, 14:54 last edited by Klaus
Oh, now I see.
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AB isn't the hypotenuse.
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wrote on 28 Dec 2020, 15:12 last edited by
Nope that was just a typo on my part when I went back to clarify!
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wrote on 28 Dec 2020, 15:17 last edited by kluurs
||AC represents the height - as the triangle is inverted for triangle CAB. 6 is the height of triangle PAB. ||
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wrote on 28 Dec 2020, 16:54 last edited by
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I was trying to say that the “really smart” kid cannot solve the problem because of the ambiguity stemming from there having three possible right triangles in the diagram: ABC, ACP, ABP. So when you say something like the hypothenuse has length 10, it’s ambiguous whether you’re referring to which hypothenuse, AB, AC, or BC.
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wrote on 28 Dec 2020, 17:57 last edited by
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Too lazy to look up the required formulas right now, but with A fixed as a right angle and CB equal to 10, height AP will be confined to a certain range. I guess it's at its maximum when angles C and B are both 45 degrees and even then AP will likely be less than 6? If that's the case, the scenario is simply impossible.
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wrote on 28 Dec 2020, 18:03 last edited by jon-nyc
I should have named the smart kid Nunatax.
The triangle is impossible as this simple diagram shows us:
(Recall that, if the angle CAB is a right angle, it must lie on the perimeter of the circle, per Euclid
The maximum height of AP is 5.
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wrote on 28 Dec 2020, 21:00 last edited by Klaus
Nice!
I wonder what that restriction (of the maximal height) looks like in non-Euclidean geometries.
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wrote on 28 Dec 2020, 21:31 last edited by
The original formula would only work if CAB was a right triangle anyway, or 1/2 a rectangle.
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I should have named the smart kid Nunatax.
The triangle is impossible as this simple diagram shows us:
(Recall that, if the angle CAB is a right angle, it must lie on the perimeter of the circle, per Euclid
The maximum height of AP is 5.
wrote on 29 Dec 2020, 15:41 last edited by@jon-nyc said in Puzzle time - the Triangle:
I should have named the smart kid Nunatax.
Correct, you should not have picked me! 555