Geometry (3rd Edition) Page 22 Question 20

[jzh200]

Question 20:

The lengths of the two shorter legs of a right triangle add up to 40 units. What is the maximum possible area of the triangle?

MGMAT Answer:

200 Square Units. You can think of a triangle as half of a rectangle. Constructing this right triangle with legs adding up to 40 is equivalent to constructing the rectangle with a perimeter of 80. Since the area of the triangle is half that of the rectangle, you can use the previously mentioned technique for maximizing the area of the rectangle: of all rectangles with a given perimeter, the square has the greatest area. The desired rectangle is thus 20 by 20 square and the right triangle has the area (1/2)(20)(20)=200 units.

My Comments:

I understand that in order to maximize the area of the rectangle we are setting it up as square. However, would we end up failing the Triangle Inequality Theorem, where the sum of any two sides of a triangle must be greater than the third side? Since this is a right triangle, wouldn’t this pose an issue considering the fact that we don’t know the third side?

[Ben Ku]

Since this is a right triangle, we know exactly the third side. According to the Pythagorean Theorem, which relates the sides in a right triangle, a^2 + b^2 = c^2. In the triangle with maximum area, the hypotenuse would be 20*sqrt(2).