Triangle

For ten years a teacher exposed his student to the following test:

A right-angle triangle has hypoteneuse 10 inches, and the height drawn from the right angle to the hypoteneuse is 6 inches. Find the area of this right-angle triangle.

For 10 years the students satisfied their teacher with the following answer: Area = (10*6)/2 = 30 square inches.

Then at the eleventh year a smart student came, who could not solve this problem. Why?

Solution

 Data (i.e. the things given):  Angle at A a right angle, AP perpendicular to BC, BC = 10,  AP = 6.

Quaesita (the things sought):  Area of the triangle ABC.

Naive approach (wrong):  Area of a triangle is half the base times the height.  In this case the base is 10, the height is 6, so the area is 30 square units.

Smart approach (correct):  Construct a circle with BC as diameter.  Plainly the radius of this circle is 5. 

By a well-known theorem of plane geometry (Euclid’s III.31), if the angle at A is to be a right angle, then A must lie on the perimeter of the circle. 

It follows that the maximum possible value of AP is 5.

The triangle as given is therefore an impossible object, and the problem has no solution.

The Monkey’s Mother

 

A rope hangs over a pulley.  On one end is a weight.  Balanced on the other end is a monkey of equal weight.  The rope weighs 4oz. per foot.  The age of the monkey and the age of its mother together equal 4 years.  The weight of the monkey is as many pounds as its mother is years old.  The mother is twice as old as the monkey was when the mother was half as old as the monkey will be when the monkey is three times as old as the mother was when the mother was three times as old as the monkey.  The weight of the weight plus the weight of the rope is half as much again as the difference between twice the weight of the weight and the weight of the monkey.  How long is the rope?