Question

(1 point) Find the maximal area of a right triangle with hypotenuse of length 3.

Answer 1

The maximal area of a right triangle with a hypotenuse of length 3 is achieved when both legs are equal due to the Pythagorean theorem, resulting in an area of approximately 2.25 square units.

Step-by-step explanation:

To find the maximal area of a right triangle with a hypotenuse of length 3, we must consider the Pythagorean theorem and how the triangle's legs contribute to its area. The Pythagorean theorem is expressed as a² + b² = c², where a and b are the triangle's legs and c is the hypotenuse. To maximize the area, we look for when the legs are equal, that is when the triangle is isosceles, since the area is maximized when the product of the legs is the greatest. Therefore, if the hypotenuse is 3 and both legs are equal, they would each be √(3²/2) according to the Pythagorean theorem. This makes each leg √4.5 or approximately 2.121. The area of the triangle would then be 1/2 × base × height or 1/2 × 2.121 × 2.121, which is approximately 2.25. Hence, the maximal area of such a triangle is roughly 2.25 square units.

Answer 2

The maximal area of a right triangle with a hypotenuse of length 3 is achieved when the triangle is isosceles, with both legs equal. Using the Pythagorean theorem and area formula, the maximal area is found to be approximately 2.25 square units.

Step-by-step explanation:

To find the maximal area of a right triangle with a given hypotenuse, we use the properties of right triangles and the formula for the area of a triangle. The Pythagorean theorem states that for a right triangle with legs of length a and b, and hypotenuse of length c, the relationship between the lengths is a² + b² = c². To maximize the area, we want to maximize the product of the legs while keeping the hypotenuse constant. This occurs when the triangle is an isosceles right triangle, meaning both legs are of equal length.

The length of each leg a is thus found by dividing the hypotenuse by √2, giving us a = √(3²/2) = √(9/2) = √4.5 ≈ 2.12. The formula for the area of a triangle, which is (1/2) × base × height, becomes (1/2) × a × a. Plugging in the value of a, we find the maximal area to be approximately (1/2) × 2.12 × 2.12 ≈ 2.25 square units.