5.4: The Pythagorean Theorem

Slides:

Advertisements

Similar presentations

7.2 Converse of Pythagorean Theorem

Advertisements

Chapter 9 Summary. Similar Right Triangles If the altitude is drawn to the hypotenuse of a right triangle, then the 3 triangles are all similar.

Geometry’s Most Elegant Theorem Pythagorean Theorem Lesson 9.4.

Lesson 10.1 The Pythagorean Theorem. The side opposite the right angle is called the hypotenuse. The other two sides are called legs. We use ‘a’ and ‘b’

Altitudes Recall that an altitude is a segment drawn from a vertex that is perpendicular to the opposite of a triangle. Every triangle has three altitudes.

8-1 The Pythagorean Theorem and Its Converse. Parts of a Right Triangle In a right triangle, the side opposite the right angle is called the hypotenuse.

8.1 Pythagorean Theorem and Its Converse

Lesson 9.4 Geometry’s Most Elegant Theorem Objective: After studying this section, you will be able to use the Pythagorean Theorem and its converse.

Section 11.6 Pythagorean Theorem. Pythagorean Theorem: In any right triangle, the square of the length of the hypotenuse equals the sum of the squares.

Geometric Mean & the Pythagorean Thm. Section 7-1 & 7-2.

+ Warm Up B. + Homework page 4 in packet + #10 1. Given 2. Theorem Given 4. Corresponding angles are congruent 5. Reflexive 6. AA Similarity 7.

MA.912.T.2.1 CHAPTER 9: RIGHT TRIANGLES AND TRIGONOMETRY.

Section 8-1: The Pythagorean Theorem and its Converse.

7.4 Similarity in Right Triangles

Section 8-1 Similarity in Right Triangles. Geometric Mean If a, b, and x are positive numbers and Then x is the geometric mean. x and x are the means.

Objective The student will be able to:

Triangles and Lines – Special Right Triangles There are two special right triangles : 30 – 60 – 90 degree right triangle 45 – 45 – 90 degree right triangle.

Right Triangles and Trigonometry Chapter Geometric Mean  Geometric mean: Ex: Find the geometric mean between 5 and 45 Ex: Find the geometric mean.

Warm Up Week 7. Section 9.1 Day 1 I will solve problems involving similar right triangles. Right Triangle – Altitude to Hypotenuse If the altitude.

Geometric Mean and the Pythagorean Theorem

Special Right Triangles

Pythagorean Theorem Unit 7 Part 1. The Pythagorean Theorem The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.

The Pythagorean Theorem

Warm Up. 9.4 Geometry’s Most Elegant Theorem Pythagorean Theorem.

Geometry Section 7.2 Use the Converse of the Pythagorean Theorem.

Geometry Chapter 8 Review. Geometric Mean Find the geometric mean between the two numbers. 7.5 and and 49.

Pythagorean Theorem and Its Converse Chapter 8 Section 1.

The Pythagorean Theorem Use the Pythagorean Theorem to find the missing measure in a right triangle including those from contextual situations.

3/11-3/ The Pythagorean Theorem. Learning Target I can use the Pythagorean Theorem to find missing sides of right triangles.

Section 8-3 The Converse of the Pythagorean Theorem.

Use Similar Right Triangles

Similar Right triangles Section 8.1. Geometric Mean The geometric mean of two numbers a and b is the positive number such that a / x = x / b, or:

Converse of the Pythagorean Theorem

Introduction to Chapter 4: Pythagorean Theorem and Its Converse

8.1 Pythagorean Theorem and Its Converse

The Pythagorean Theorem

Pythagorean Theorem and it’s Converse

Pythagorean Theorem and Special Right Triangles

Section 10.2 Triangles Triangle: A closed geometric figure that has three sides, all of which lie on a flat surface or plane. Closed geometric figures.

The Converse of the Pythagorean Theorem

Geometric Mean Pythagorean Theorem Special Right Triangles

Section 7.2 Pythagorean Theorem and its Converse Objective: Students will be able to use the Pythagorean Theorem and its Converse. Warm up Theorem 7-4.

Pythagorean Theorem and Its Converse

CHAPTER 8 Right Triangles.

Math 3-4: The Pythagorean Theorem

Section 5.5: Special Right Triangles

4.3 The Rectangle, Square and Rhombus The Rectangle

Chapter 7.3 Notes: Use Similar Right Triangles

8-2 The Pythagorean Theorem and Its Converse

8.1 Pythagorean Theorem and Its Converse

5.7: THE PYTHAGOREAN THEOREM (REVIEW) AND DISTANCE FORMULA

4.3 The Rectangle, Square and Rhombus The Rectangle

Pythagorean Theorem a²+ b²=c².

7-1 and 7-2: Apply the Pythagorean Theorem

8.1 Pythagorean Theorem and Its Converse

C = 10 c = 5.

Right Triangles Unit 4 Vocabulary.

Similar Right Triangles

5.4: The Pythagorean Theorem

Y. Davis Geometry Notes Chapter 8.

Geometric Mean Pythagorean Theorem Special Right Triangles

8.1 Geometric Mean The geometric mean between two numbers is the positive square root of their product. Another way to look at it… The geometric mean is.

Using Similar Right Triangles

Geometric Mean and the Pythagorean Theorem

Similar Right Triangles

Section 8.1 – 8.2 Geometric Mean Pythagorean Theorem

Converse to the Pythagorean Theorem

7-2 PYTHAGOREAN THEOREM AND ITS CONVERSE

Section 3.3 Isosceles Triangles

Presentation transcript:

5.4: The Pythagorean Theorem Theorem 5.4.1: The altitude drawn to the hypotenuse of a right triangle separates the right triangle into two right triangles that are similar to each other and to the original triangle. In the figure below, AD and DB are segments of the hypotenuse and AD is the segment adjacent to leg AC and DB is the segment adjacent to leg BC. 12/3/2018 Section 5.4 Nack

Theorem 5.4.2: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse. AD = CD CD DB Note: CD is a geometric mean because the second and the third terms are identical. Lemma 5.4.3: The length of each leg of a right triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg. 12/3/2018 Section 5.4 Nack

Pythagorean Theorem Theorem 5.4.4: The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the legs. Proof p. 243: From diagram (b) c = b and c = a b x a y Cross multiply: b² = cx and a²= cy Use the Addition Property of Equality: a² + b² = cx + cy =c (x + y) But x + y = c so: a² + b² = c² 12/3/2018 Section 5.4 Nack

Use the Pythagorean Theorem p. 246 Ex. 3 p. 244 Theorem 5.4.5: (Converse of Pythagorean Theorem): If a, b, and c are the lengths of the three sides of a triangle, with c the length of the longest side, and if c² = a² + b², then the triangle is a right triangle with the right angle opposite the side of length c. Proof of HL (Theorem 5.4.6): If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of the second right triangle, then the triangles are congruent. Use the Pythagorean Theorem p. 246 Ex. 3 p. 244 12/3/2018 Section 5.4 Nack

Pythagorean Triples A set of three natural numbers (a, b, c) for which See Table 5.1 p. 247 Theorem 5.4.7: Let a, b, and c represent the lengths of the three sides of a triangle, with c the length of the longest side. If c² > a² + b² , then the triangle is obtuse and the obtuse angle lies opposite the side of length c. If c² < a² + b² , then the triangle is acute. Ex. 6 p. 247 12/3/2018 Section 5.4 Nack