Download 8.2 Use Properties of Parallelograms

Heron
• Heron of Alexandria (c.
10–70 AD) was an
ancient Greek
mathematician and
engineer. He is
considered the greatest
experimenter of antiquity
and his work is
representative of the
Hellenistic scientific
tradition.
Heron’s Formula
• Find the area of a triangle in terms of the
lengths of its sides 𝑎, 𝑏, and 𝑐.
𝐴=
𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
Where
1
𝑠 = (𝑎 + 𝑏 + 𝑐)
2
Example
• Use Heron’s formula to find the area of
each triangle.
𝑎 = 8 𝑖𝑛, 𝑏 = 9 𝑖𝑛, 𝑐 = 10 𝑖𝑛
Example
• Find the area using Herons formula
Polygon Area Formulas
Parallelograms
What makes a polygon a parallelogram?
10.1 Parallelograms
Objectives:
1. To discover and use properties of
parallelograms
2. To find side, angle, and diagonal
measures of parallelograms
3. To find the area of parallelograms
Parallelogram
A parallelogram is a
quadrilateral with
both pairs of
opposite sides
parallel.
• Written PQRS
• PQ||RS and QR||PS
Theorem 1
If a quadrilateral is a
parallelogram, then
its opposite sides
are congruent.
If PQRS is a parallelogram, then PQ  RS and QR  PS.
Theorem 2
If a quadrilateral is a
parallelogram, then
its opposite angles
are congruent.
If PQRS is a parallelogram, then P  R and Q  S .
Theorem 3
If a quadrilateral is a
parallelogram, then
consecutive angles
are supplementary.
If PQRS is a parallelogram, then x + y = 180°.
Theorem 4
If a quadrilateral is a
parallelogram, then
its diagonals bisect
each other.
Example
The diagonals of
parallelogram
LMNO
intersect at
point P. What
are the
coordinates of
P?
Bases and Heights
Any one of the sides of a parallelogram can
be considered a base. But the height of a
parallelogram is not necessarily the length
of a side.
Bases and Heights
The altitude is any segment from one side of
the parallelogram perpendicular to a line
through the opposite side. The length of
the altitude is the height.
Bases and Heights
The altitude is any segment from one side of
the parallelogram perpendicular to a line
through the opposite side. The length of
the altitude is the height.
Area of a Parallelogram Theorem
The area of a parallelogram is the product of
a base and its corresponding height.
A = bh
Height (h)
Height (h)
Base (b)
Base (b)
Area of a Parallelogram Theorem
The area of a parallelogram is the product of
a base and its corresponding height.
A = bh
Example
Find the area of parallelogram PQRS.
Example
What is the height of
a parallelogram that
has an area of 7.13
m2 and a base 2.3
m long?
Example
Find the area of each triangle or parallelogram.
1.
2.
3.
Example
Find the area of the parallelogram.
10.2 Rhombuses (Kites), Rectangles, and
Squares
Objectives:
1. To discover and use properties of
rhombuses, rectangles, and squares
2. To find the area of rhombuses, kites
rectangles, and squares
Example 2
Below is a concept map showing the
relationships between some members of the
parallelogram family. This type of concept
map is known as a Venn Diagram. Fill in the
missing names.
Example 2
Below is a concept map showing the
relationships between some members of the
parallelogram family. This type of concept
map is known as a Venn Diagram.
Example 5
Classify the special quadrilateral. Explain
your reasoning.
Diagonal Theorem 1
A parallelogram is a rectangle if and only if
its diagonals are congruent.
Example
You’ve just had a new door installed, but it
doesn’t seem to fit into the door jamb
properly. What could you do to determine
if your new door is rectangular?
Diagonal Theorem 2
A parallelogram is a rhombus if and only if its
diagonals are perpendicular.
Rhombus Area
Since a rhombus is
a parallelogram, we
could find its area
by multiplying the
base and the
height.
A  bh
Rhombus/Kite Area
However, you’re not
always given the
base and height, so
let’s look at the two
diagonals. Notice
that d1 divides the
rhombus into 2
congruent triangles.
Ah, there’s a couple of
triangles in there.
1
A  bh
2
Rhombus/Kite Area
So find the area of
one triangle, and
then double the
result.
1

A  2 b  h
2

1 
1
A  2  d1  d 2 
2 
2
1
1

A  2  d1  d 2   d1  d 2
2
4

Ah, there’s a couple of
triangles in there.
1
A  bh
2
1
A  d1  d 2
2
Polygon Area Formulas
Exercise 11
Find the area of the shaded region.
1.
2.
3.
Trapezoids
What makes a quadrilateral a trapezoid?
Trapezoids
A trapezoid is a
quadrilateral with
exactly one pair of
parallel opposite
sides.
Polygon Area Formulas
Trapezoid Parts
• The parallel sides
are called bases
• The non-parallel
sides are called
legs
• A trapezoid has two
pairs of base angles
Trapezoids and Kites
Objectives:
1. To discover and use properties of
trapezoids and kites.
Example 1
Find the value of x.
C
B
100
x
A
D
Trapezoid Theorem 1
If a quadrilateral is a trapezoid, then the
consecutive angles between the bases are
supplementary.
C
B
y
x
A
t
r
D
If ABCD is a trapezoid, then x + y = 180° and r + t = 180°.
Isosceles Trapezoid
An isosceles trapezoid is a trapezoid with
congruent legs.
Trapezoid Theorem 2
If a trapezoid is isosceles, then each pair of
base angles is congruent.
Trapezoid Theorem 3
A trapezoid is isosceles if and only if its
diagonals are congruent.
T
i
Example 2
Find the measure of each missing angle.
Kites
What makes a quadrilateral a kite?
Kites
A kite is a
quadrilateral that
has two pairs of
consecutive
congruent sides,
but opposite sides
are not congruent.
Angles of a Kite
You can construct a kite by joining two
different isosceles triangles with a common
base and then by removing that common
base.
Two isosceles triangles can form one kite.
Angles of a Kite
Just as in an
isosceles triangle,
the angles between
each pair of
congruent sides are
vertex angles. The
other pair of angles
are nonvertex
angles.
Kite Theorem 1
If a quadrilateral is a kite, then the nonvertex
angles are congruent.
Kite Theorem 2
If a quadrilateral is a kite, then the diagonal
connecting the vertex angles is the
perpendicular bisector of the other
C
diagonal.
B
E
A
and CE  AE.
D
Kite Theorem 3
If a quadrilateral is a kite, then a diagonal
bisects the opposite non-congruent vertex
C
angles.
D
B
A
If ABCD is a kite, then BD bisects B and D.
Example 5
Quadrilateral DEFG is
a kite. Find mD.
Example 6
Find the area of kite PQRS.