Question

Multiple Choice What is the maximum area of a right triangle with hypotenuse 10? \(\begin{array}{llll}{\text { (A) } 24} & {\text { (B) } 25} & {\text { (C) } 25 \sqrt{2}} & {\text { (D) } 48} & {\text { (E) } 50}\end{array}\)

Short answer

The maximum area of a right triangle with hypotenuse 10 is 25 square units. So, option (B) is the correct answer.

Step-by-step solution

Finding the length of the sides

Firstly, it can be noticed that for the area of a right triangle to be maximum, base and height should be of the same length because in this way the product will be maximized. Therefore, assuming a right triangle with the same base and height, we use Pythagorean theorem to find the lengths of these sides. Thus, if \( a \) is the length of the base and the height, \( a^2 + a^2 = 10^2 \) or \( 2a^2 = 100 \) or \( a^2 = 50 \). Hence, \( a = \sqrt{50} \).

Calculating the maximum area

With both base and height equal to \( \sqrt{50} \), the area, \( A \), of the right triangle can be found by using the formula \( A = 1/2 \times \text{base} \times \text{height} = 1/2 \times \sqrt{50} \times \sqrt{50} = 1/2 \times 50 = 25 \).