Algebra Class: Simplifying Radicals

Study Notes

Radicals are expressions with roots, such as square roots, cube roots, etc. Simplifying them involves reducing the expression to its simplest form.
Prime Factorization: Find the prime factors of the number under the radical (e.g., the radicand).
Perfect Roots: Identify perfect squares, cubes, etc., within the prime factorization.
Rule of Radicals: The square root of a product is the product of the square roots (√(ab) = √a * √b). The square root of a quotient is the quotient of the square roots (√(a/b) = √a / √b).
Example: Simplify √18. The prime factorization of 18 is 2 * 3 * 3. √18 = √(2 * 3 * 3) = √(2) * √(3 * 3) = √2 * 3 = 3√2
Simplifying with Variables: Follow the same prime factorization and perfect root rules. A variable under a square root needs to have an even power to be removed from under the root. Example: √x² = x (since x² implies two factors, x multiplied by x). √x³ = x√x (one x can be removed).

Definition: An angle bisector is a ray that divides an angle into two congruent angles.
Theorem: An angle bisector in a triangle divides the opposite side into segments proportional to the lengths of the other two sides.
Formula: If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the lengths of the adjacent sides. If AD bisects ∠BAC, then AB/AC = BD/DC.
Key Concepts: The ratios of corresponding sides are equal.
Proof: The proof of the Angle Bisector Theorem often involves constructing parallel lines to a side of the triangle and referencing the properties of similar triangles. Similarity is a key property in the proof process.
Solving Problems: Angle bisector theorems are used to find unknown side lengths or distances in triangles when an angle is bisected.
Example: If AB = 6, AC = 8, and BD = 4, find DC. Using the formula, 6/8 = 4/DC. Solving this proportion gives DC = (8 * 4) / 6 = 32/6 = 16/3.
Relationship with similar triangles: The angle bisector creates two similar triangles. This is crucial for proofs and for setting up the proportions used in the theorem.
Application: Angle bisectors are used in many geometric constructions and problem solving, often in conjunction with other geometric theorems. Knowing the Angle Bisector theorem allows problems that involve angle measurements to now work with side lengths.