Toughest question asked in the Microsoft Interview!!!

This question was asked to one of NIT student in his campus placement interview with Microsoft.

Interviewer: Ok, so one last question. A right triangle has a hypotenuse equal to 10 and an altitude to the hypotenuse equal to 6. Find the area of the triangle.

 The guy started thinking 'Why would a software company ask a geometry question and that too such a trivial one! Maybe it is a trick question!? Maybe it isn't a trick question and he just wants me to think otherwise so that I would screw up even this paltry question!?' 

He contemplated for a while and answered:

Student : Sir, as area of any  triangle is 0.5*base*height, the answer to this question would be 0.5 *10*6 which evaluates to 30!

Interviewer: Are you sure? Think about it again!

*My friend thought for a while and replied with full confidence*

Student: Yes sir, I am sure the area of triangle is 30. You are just messing up with my brain to make me think otherwise so that I would commit error even in this trivial question.

(This was the exact response which my friend gave to the interviewer!)

Interviewer: Well, your answer is wrong XYZ, and that was the last question of this interview. You can wait outside until we declare the results.

Student: Sir, could you please tell me what's the correct answer?

Interviewer: The correct answer is, such a triangle cannot exist. If you think about it, you will come to know why!

And that student left the room dumbfounded, still thinking about why that triangle cannot exist.

Verdict: He wasn't selected!

Here's the solution: It turns out that the maximum length of the altitude to hypotenuse in the above triangle can only be 5 and not 6, so its maximal area would be 25.

The angle opposite the hypotenuse must be a right angle of 90 degrees. This means the two sides of the triangle must subtend a 180 degree angle in a circle. The hypotenuse must be the diameter of a circle, and the third point can be any point on the circle (except the endpoints of the hypotenuse).

The vertical distance from the third point to the hypotenuse is the altitude to the hypotenuse. This is largest when the third point is at the top or bottom of the circle, and the vertical distance is equal to the radius of the circle (half the length of the hypotenuse, which is the diameter of the circle).

Therefore, a right triangle with a hypotenuse of 10 can have an altitude on its hypotenuse of at most 5.

I would say it was one of the most difficult questions to answer because the hard question was very well disguised in a trivial question. And although it seemed quite strange that a software giant like Microsoft asked a geometry question, it wasn't strange at all. The real motive could have been to check whether the candidate has good analytical skills and levelheadedness, both of which are quite essential in the programming world.