Question

The legs of a right triangle have lengths A and B satisfying A + B = 10. Which values of A and B maximize the area of the triangle?

Answer 1

Solution:

We are going to find the maximum value of Area of Triangle.

Given,

The legs of a right triangle have lengths A and B

$$A + B = 10$$

Here Base = A and Height = B

We have a formula for area of the triangle, that is

Area of the Right triangle =$\frac {1} {2} \times Base \times height$

= $\frac {1} {2} \times A \times B$

Now we can solvve for A and B using the concept A.M (Arithmetic mean) and G.M (Geometric mean)

$$A.M \space of \space A \space and \space B = \frac {A+B} {2}$$

$$G.M \space of \space A \space and \space B = \sqrt {AB}$$

We know,

$$G.M \le A.M$$

$$\sqrt {AB}\le \frac {A+B} {2}$$

$$\sqrt {AB}\le \frac {10} {2} \\\sqrt {AB}\le 5$$

$$AB \le 25$$

Maximum Area would be = $\frac {1}{2} AB = 12.5$, if the A and B are equal

So, A = B = 5.

Answer: A = B = 5