Example 1 What is the area of a square with a side length of 4 inches? x inches? What is the side length of a square with an area of 25 in2? x in2? x

Parts of a Right Triangle Which segment is the longest in any right triangle?

Apply the Pythagorean Theorem Objectives: To discover and use the Pythagorean Theorem To use Pythagorean Triples to find quickly find a missing side length in a right triangle

The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Example 2 The triangle below is definitely not a right triangle. Does the Pythagorean Theorem work on it?

Example 3 How high up on the wall will a twenty-foot ladder reach if the foot of the ladder is placed five feet from the wall?

Example 4: SAT In figure shown, what is the length of RS?

Example 5 What is the area of the large square?

Example 6 Find the area of the triangle.

Pythagorean Triples Three whole numbers that work in the Pythagorean formulas are called Pythagorean Triples.

Example 7 What happens if you add the same length to each side of a right triangle? Do you still get another right triangle?

Example 8 What happens if you multiply all the side lengths of a right triangle by the same number? Do you get another right triangle?

Pythagorean Multiples Pythagorean Multiples Conjecture: If you multiply the lengths of all three sides of any right triangle by the same number, then the resulting triangle is a right triangle. In other words, if a2 + b2 = c2, then (an)2 + (bn)2 = (cn)2.

Pythagorean Triples

Primitive Pythagorean Triples A set of Pythagorean triples is considered a primitive Pythagorean triple if the numbers are relatively prime; that is, if they have no common factors other than 1. 3-4-5 5-12-13 7-24-25 8-15-17 9-40-41 11-60-61 12-35-37 13-84-85 16-63-65 20-21-29 28-45-53 33-56-65 36-77-85 39-80-89 48-55-73 65-72-97

Example 9 Find the length of one leg of a right triangle with a hypotenuse of 35 cm and a leg of 28 cm.

Example 10 Use Pythagorean Triples to find each missing side length.

Example 11 A 25-foot ladder is placed against a building. The bottom of the ladder is 7 feet from the building. If the top of the ladder slips down 4 feet, how many feet will the bottom slip out?

Converse of the Pythagorean Theorem Objectives: To investigate and use the Converse of the Pythagorean Theorem To classify triangles when the Pythagorean formula is not satisfied

7.2 Converse of the Pythagorean Theorem Converse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then it is a right triangle.

Example 3 Which of the following is a right triangle?

Example 4 Tell whether a triangle with the given side lengths is a right triangle. 5, 6, 7 5, 6, 5, 6, 8

These numbers are called the Perfect Squares. Example 1 Write the first 20 terms of the following sequence: 1, 4, 9, 16, … x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x2 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 These numbers are called the Perfect Squares.

Square Roots The number r is a square root of x if r2 = x. This is usually written Any positive number has two real square roots, one positive and one negative, √x and -√x √4 = 2 and -2, since 22 = 4 and (-2)2 = 4 The positive square root is considered the principal square root

Example 2 Use a calculator to evaluate the following:

Example 3 Use a calculator to evaluate the following:

Properties of Square Roots Properties of Square Roots (a, b > 0) Product Property Quotient Property

7.4 Special Right Triangles Simplifying Radicals Objectives: To simplify square roots To solve quadratic equations

Simplifying Square Root The properties of square roots allow us to simplify radical expressions. A radical expression is in simplest form when: The radicand has no perfect-square factor other than 1 There’s no radical in the denominator

Simplest Radical Form Like the number 3/6, is not in its simplest form. Also, the process of simplification for both numbers involves factors. Method 1: Factoring out a perfect square.

Simplest Radical Form In the second method, pairs of factors come out of the radical as single factors, but single factors stay within the radical. Method 2: Making a factor tree.

Simplest Radical Form This method works because pairs of factors are really perfect squares. So 5·5 is 52, the square root of which is 5. Method 2: Making a factor tree.

Investigation 1 Express each square root in its simplest form by factoring out a perfect square or by using a factor tree.

Exercise 4a Simplify the expression.

Exercise 4b Simplify the expression.

Example 5 Simplify the expression.

Example 6 Evaluate, and then classify the product. (√5)(√5) = (2 + √5)(2 – √5) =

Conjugates are Magic! The radical expressions a + √b and a – √b are called conjugates. The product of two conjugates is always a rational number

Example 7 Identify the conjugate of each of the following radical expressions: √7 5 – √11 √13 + 9

Example 8 Recall that a radical expression is not in simplest form if it has a radical in the denominator. How could we use conjugates to get rid of any damnable denominator-bound radicals?

Rationalizing the Denominator We can use conjugates to get rid of radicals in the denominator: The process of multiplying the top and bottom of a radical expression by the conjugate of the denominator is called rationalizing the denominator. Fancy One

Exercise 9a Simplify the expression.

Exercise 9b Simplify the expression.

Solving Quadratics If a quadratic equation has no linear term, you can use square roots to solve it. By definition, if x2 = c, then x = √c and x = −√c, usually written x = ±√c You would only solve a quadratic by finding a square root if it is of the form ax2 = c In this lesson, c > 0, but that does not have to be true.

Solving Quadratics If a quadratic equation has no linear term, you can use square roots to solve it. By definition, if x2 = c, then x = √c and x = -√c, usually written x = √c To solve a quadratic equation using square roots: Isolate the squared term Take the square root of both sides

Exercise 10a Solve 2x2 – 15 = 35 for x.

Exercise 10b Solve for x.

The Quadratic Formula Let a, b, and c be real numbers, with a ≠ 0. The solutions to the quadratic equation ax2 + bx + c = 0 are Song 1: Song 2:

Exercise 11a Solve using the quadratic formula. x2 – 5x = 7

Exercise 11b Solve using the quadratic formula. x2 = 6x – 4 4x2 – 10x = 2x – 9 7x – 5x2 – 4 = 2x + 3

Exercise 12 Based on the previous Exercise, How can the quadratic formula tell you how many solutions to expect? How can the quadratic formula tell you what kind of solutions to expect: Real or imaginary, rational or irrational? How are the roots related to each other if they are irrational or imaginary?

The Discriminant In the quadratic formula, the expression b2 – 4ac is called the discriminant. Discriminant