1. MVUSD GeoMetry CH-5 Instructor: Leon Robert Manuel PRENTICE HALL MATHEMATICS: MEASURING IN THE PLANE In this chapter students will:  build on their knowledge of triangles and quadrilaterals by learning to find the area and perimeter of parallelograms, triangles, trapezoids, and regular polygons  study the Pythagorean Theorem, its converse, and the properties of 30°-60°-90° triangles  learn to find circumference, arc length, and area of circles, sectors of circles, and segments of circles  GOTO P-81 -87

2. Chapter 5 Section 5.03 Instructor: Leon Robert Manuel The Pythagorean Theorem and Its Converse. CA Geometry STD: 10, 6

3. End of Lecture / Start of Lecture mark

4. The Pythagorean Theorem. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. A B C b a c

5. Pythagorean theorem Proof A proof of Pythagorean theorem is clear. Consider a right-angled triangle ABC with legs a, b and a hypotenuse c.

6. Pythagorean theorem Build the square AKMB, using hypotenuse AB as its side. Then continue sides of the right- angled triangle ABC so, to receive the square CDEF, the side length of which is equal to a + b.

7. Pythagorean theorem Now it is clear, that an area of the square CDEF is equal to ( a + b )². On the other hand, this area is equal to a sum of areas of four right-angled triangles and a square AKMB, that is (a + b)² = c² + 4 (ab / 2) a² + 2ab + b² = c² + 2ab, - 2ab -2ab a² + b² = c²

8. Relation of sides. In general case ( for any triangle ) we have: c² = a² + b² – 2ab · cos C o, where C – an angle between sides a and b. A B C a b

9. The Pythagorean Theorem. When the lengths of the sides of a right triangle are integers, the integers form a Pythagorean Triple. 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

10. Converse of Pythagorean Theorem. If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. b a c

11. Converse of Pythagorean Theorem. GEOMETRY LESSON 5-7

12. Find the Area and Perimeter Measuring in Plane: Circles Circumference & Arc Length GEOMETRY LESSON 5-7