A right triangle has hypotenuse of length 65 . a. Let a and...

A right triangle has hypotenuse of length 65 .
a. Let $a$ and $b$ denote the lengths of the legs. Use the fact that the hypotenuse is 65 to express $b$ in terms of $a$.
b. Express the area in terms of $a$ alone.
c. Find the lengths of the legs giving maximum area.

Key Concepts

Pythagorean Theorem

The Pythagorean theorem establishes the relationship between the three sides of a right triangle by stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This fundamental geometric principle allows us to express one leg in terms of the other when the length of the hypotenuse is known.

Expressing Variables in Terms of Each Other

When faced with a constraint involving two variables, such as the relationship between the legs of a triangle given by the Pythagorean theorem, one variable can be rewritten in terms of the other. This substitution simplifies the problem by reducing the number of variables, making it easier to analyze and optimize functions like area.

Area of a Right Triangle

The area of a right triangle is given by one-half the product of its two perpendicular legs. This formula is central when the goal is to maximize the area, as it sets up the objective function to be optimized in terms of one variable once a substitution is made from the Pythagorean relationship.

Optimization in Geometry

Optimization in a geometric context involves determining the maximum or minimum value of a geometric quantity, such as area, subject to specific constraints. This typically includes forming an objective function that incorporates the constraints and then using techniques like calculus (derivatives) or algebraic methods (completing the square) to identify the optimal solution.

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