heron

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

ParallelogramsWhat makes a polygon a parallelogram?

10.1 Parallelograms

Objectives:1. To discover and use properties of

parallelograms2. To find side, angle, and diagonal

measures of parallelograms3. To find the area of parallelograms

ParallelogramA parallelogram is a

quadrilateral with both pairs of opposite sides parallel.

• Written PQRS • PQ||RS and QR||PS

Theorem 1If a quadrilateral is a

parallelogram, then its opposite sides are congruent.

If PQRS is a parallelogram, then and . RSPQ PSQR

Theorem 2If a quadrilateral is a

parallelogram, then its opposite angles are congruent.

If PQRS is a parallelogram, then and .

RP SQ

Theorem 3If a quadrilateral is a

parallelogram, then consecutive angles are supplementary.

If PQRS is a parallelogram, then x + y = 180°.

Theorem 4If 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 HeightsAny 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 HeightsThe 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 TheoremThe 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 TheoremThe 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 squares2. To find the area of rhombuses, kites

rectangles, and squares

Example 2Below 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 2Below 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 5Classify the special quadrilateral. Explain

your reasoning.

Diagonal Theorem 1A 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 2A 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.

12

A b h

Rhombus/Kite AreaSo find the area of

one triangle, and then double the result.

12

A b h

122

A b h

1 21 122 2

A d d

1 2124

A d d

1 212

d d 1 2

12

A d d

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

Polygon Area Formulas

Exercise 11Find the area of the shaded region.1. 2. 3.

TrapezoidsWhat makes a quadrilateral a trapezoid?

TrapezoidsA 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 1Find the value of x.

100

xA D

B C

Trapezoid Theorem 1If 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 TrapezoidAn isosceles trapezoid is a trapezoid with

congruent legs.

Trapezoid Theorem 2If a trapezoid is isosceles, then each pair of

base angles is congruent.

Trapezoid Theorem 3A trapezoid is isosceles if and only if its

diagonals are congruent.

Ti

Example 2Find the measure of each missing angle.

KitesWhat makes a quadrilateral a kite?

KitesA kite is a

quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Angles of a KiteYou 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 KiteJust 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 1If a quadrilateral is a kite, then the nonvertex

angles are congruent.

Kite Theorem 2If 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 3If 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 5Quadrilateral DEFG is

a kite. Find mD.

Example 6Find the area of kite PQRS.