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MVUSD GeoMetry CH-5
Instructor: Leon Robert Manuel
PRENTICE HALL MATHEMATICS: MEASURING IN THE PLANE
In this chapter students will:
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
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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
CA Geometry STD: 10, 6
The Pythagorean Theorem and Its Converse.
End of Lecture / Start of Lecture mark
b  b  4ac
x
2a
2
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
2
2
2
A
b
c
C
a
B
Pythagorean theorem Proof
a b  c
2
2
A proof of Pythagorean theorem is clear. Consider a
right-angled triangle ABC with legs a, b and a hypotenuse c.
2
Pythagorean theorem
Build the square AKMB, using hypotenuse AB
as its side. Then continue sides of the rightangled 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
b
C
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.
a b  c
2
2
2
b
c
a
Converse of Pythagorean Theorem.
GEOMETRY LESSON 5-7
Measuring in Plane: Circles Circumference & Arc Length
GEOMETRY LESSON 5-7
Find the Area and Perimeter