Silly standardized test question puzzler

Tanya Khovanova posed an interesting puzzle from VI Arnold:

I have a tiny book written by Vladimir Arnold Problems for Kids from 5 to 15. A free online version of this book is available in Russian. The book contains 79 problems, and problem Number 6 criticizes American math education. Here is the translation:

(From an American standardized test) A hypotenuse of a right triangle is 10 inches, and the altitude having the hypotenuse as its base is 6 inches. Find the area of the triangle. American students solved this problem successfully for 10 years, by providing the “correct” answer: 30 inches squared. However, when Russian students from Moscow tried to solve it, none of them “succeeded”. Why?

This one is worth thinking about.

The Russian students to whom Professor Arnold posed this problem were probably members of math circles, which originated in Eastern Europe and Russia and encourage outside-the-box thinking rather than cookbook approaches to problems.
Why couldn't the Russian students solve this apparently straightforward and simple problem?

Scroll down this page for the explanation.





















The Russian students realized that the triangle specified in the problem can't exist. The altitude to the hypotenuse of a right triangle can never be more than half the length of the hypotenuse. In fact, the altitude to the hypotenuse of a right triangle will always be exactly half the length of the hypotenuse. This fact is easy to see if you recall that every right triangle can be inscribed in a semicircle.

It's troubling to think that the American standardized testing company allowed this nonsensical problem to stay on their test for ten years!