A right triangle has hypotenuse of length 8 . Let a and b...

A right triangle has hypotenuse of length 8 . Let $a$ and $b$ denote the lengths of the legs.
a. Use the fact that the hypotenuse is 8 to find $b$ in terms of $a$.
b. Express the perimeter in terms of $a$ alone.
c. Assume now that the perimeter is 17. Find the lengths of the legs.

Key Concepts

Solving Equations with Radicals

When expressions involve square roots, such as when isolating a variable using the Pythagorean theorem, one must often solve equations that include radicals. This typically involves isolating the square root term and then squaring both sides of the equation, while taking care to consider extraneous solutions.

Pythagorean Theorem

This fundamental theorem relates the three sides of a right triangle by asserting that the square of the hypotenuse equals the sum of the squares of the two legs. It provides a way to express one leg in terms of the other when the hypotenuse is known, and it is essential for solving problems involving right triangles.

Expressing Variables in Terms of Others

In many problems, particularly in geometry, one variables is often expressed in terms of another by rearranging known formulas. This technique is crucial for reducing the number of unknowns, which facilitates further analysis or substitution into another equation.

Triangle Perimeter

The perimeter of a geometric figure is the total length around it, which in the case of a triangle is the sum of its three side lengths. Expressing the perimeter algebraically in terms of one variable can introduce additional relationships between the sides under given constraints.

Quadratic Equations

Quadratic equations arise naturally in geometric problems when variables related by the Pythagorean theorem and perimeter constraints lead to equations involving squared terms. Solving these equations, typically via the quadratic formula, is essential for finding the unknown values.

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