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 and .

Where

Example

• Use Heron’s formula to find the area of each triangle.

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 and . RSPQ PSQR

Theorem 2

If a quadrilateral is a parallelogram, then its opposite angles are congruent.

If PQRS is a parallelogram, then and .

RP SQ

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.

Base (b)

Height (h)

Base (b)

Height (h)

A = bh

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

2A b h

Rhombus/Kite Area

So find the area of one triangle, and then double the result.

1

2A b h

12

2A b h

1 2

1 12

2 2A d d

1 2

12

4A d d

1 2

1

2d d

1 2

1

2A d d

Ah, there’s a couple of triangles in there.

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.

100

xA D

B C

Trapezoid Theorem 1

If a quadrilateral is a trapezoid, then the consecutive angles between the bases are supplementary.

r

ty

xA D

B C

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.

Ti

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 diagonal.

E

A

B

C

D

and CE AE.

Kite Theorem 3

If a quadrilateral is a kite, then a diagonal bisects the opposite non-congruent vertex angles.

A

B

C

D

If ABCD is a kite, then BD bisects B and D.

Example 5

Quadrilateral DEFG is a kite. Find mD.

Example 6

Find the area of kite PQRS.