Olympiad specials

Geometry Level 4

A right triangle has a hypotenuse of length 100cm and the length of the altitude dropped to the hypotenuse is 60cm . What is this triangle's area?

The question is flawed 30000mm^{2} 30m^{2} 3000cm^{2} All of the above

Kartik Agarwal by Kartik Agarwal , 20, India

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2 solutions

Shaun Leong
Jan 16, 2016

If such a triangle exists, we have 10000 = c 2 = a 2 + b 2 ≥ 2 a b = 4 ( 1 2 a b ) = 12000 10000=c^2=a^2+b^2 \geq 2ab=4 (\frac {1}{2}ab)=12000 This is a contradiction which follows from the AM-GM Inequality.

Hence no such triangle exists.

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Indeed. Since the hypotenuse of a right triangle corresponds to a diameter of its circumcircle, the altitude to the hypotenuse cannot exceed the radius of the circimcircle, i.e., half the length of the hypotenuse.

Brian Charlesworth - 5 years, 5 months ago

A triangle inscribed in a circle where one side is the diameter of the circle is a right triangle. Based from the diagram above, the maximum possible altitude to the hypotenuse is 50 50 .

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