A right triangle has hypotenuse of length 15 . Let a and b...

A right triangle has hypotenuse of length 15 . Let $a$ and $b$ denote the lengths of the legs.
a. Use the fact that the hypotenuse is 15 to find $b$ in terms of $a$.
b. Express the perimeter in terms of $a$ alone.
c. Assume now that the perimeter is 35 . Find the lengths of the legs.

Key Concepts

Right Triangle

A right triangle is a type of triangle that includes one 90-degree angle. This geometric shape is fundamental in trigonometry and Euclidean geometry, where the relationships among its three sides are established through properties unique to right angles. Understanding a right triangle is essential for applying various mathematical theorems, such as the Pythagorean theorem, to derive relationships between its sides.

Pythagorean Theorem

The Pythagorean theorem is a key concept in geometry that relates the lengths of the sides of a right triangle. It states that the sum of the squares of the two legs is equal to the square of the hypotenuse. This theorem is widely used to express one side in terms of another and to solve problems involving distances and measurements in various fields of mathematics and applied sciences.

Expressing Variables in Terms of Another

Expressing one variable in terms of another involves using given relationships to rewrite an equation with a singular independent variable. This technique simplifies the problem by reducing the number of variables, making it easier to perform further algebraic manipulations, such as finding perimeters or areas, and ultimately solving for unknown quantities.

Perimeter

The perimeter of a polygon is the total length around the shape, calculated by summing the lengths of all its sides. In the context of triangles, the perimeter is the sum of its three side lengths. Being able to express the perimeter in terms of one variable is a valuable skill in both pure and applied mathematics, as it allows for setting up equations that model real-world problems.

Quadratic Equations

Quadratic equations are polynomial equations of degree two and are central to algebra. They often arise when solving geometric problems that involve squaring terms, such as when applying the Pythagorean theorem. Solving these equations can yield one or more solutions, providing the necessary values of variables that satisfy the conditions of the problem. Techniques for solving quadratic equations include factoring, completing the square, and using the quadratic formula.

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