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

Chapter 5 Section 5.03 Instructor: Leon Robert Manuel

The Pythagorean Theorem and Its Converse.

CA Geometry STD: 10, 6

End of Lecture / Start of Lecture mark

2 4

2

b b acx

a

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.2 2 2a b c A

BC

b

a

c

Pythagorean theorem Proof 2 2 2a b c

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

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 .

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

Relation of sides.

In general case ( for any triangle ) we have:

c² = a² + b² – 2ab · cos Co ,

where C – an angle between sides  a  and  b .

A

BC a

b

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

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.

2 2 2a b c b

a

c

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

Find the Area and Perimeter

Measuring in Plane: Circles Circumference & Arc LengthGEOMETRY LESSON 5-7GEOMETRY LESSON 5-7