Areas of Parallelograms and Triangles

Areas of Parallelograms and Triangles

Lesson 7-1

Thm 7-1 Area of a Rectangle

For a rectangle, A=bh.(Area = base · height)

h

b

AREA OF A PARALLELOGRAM

To do this let’s cut the left triangle and…

h

b

slide it…

h

h b

slide it…

h

h

b

slide it…

h

h

b

slide it…

h

hb

…thus, changing it to a rectangle.

What is the area of the rectangle?

h

b

Thm 7-2Area of a Parallelogram

For a parallelogram, A=bh.

h

b

Parts of a Parallelogram

Base – any side of the parallelogram. Altitiude – the perpendicular segment

form the line containing one base to the opposite base.

Height – length of the altitude.

Finding the Area of a Parallelogram

Find the area of the parallelogram.

A = 96m2

Finding a Missing Dimension

For parallelogram ABCD, find CF to the nearest tenth.

10 in.

12 in.13 in.

A BE

CD

F

X1st: Find area of ABCD

a = b ha = 10 (12) = 120 in2

2nd: Use area formula for other base and height

a = b h120 = 13 (x)x 9.2

Thm 7-3Area of a Triangle

For a triangle, A= ½ bh.

h

b

Finding the Area of a Triangle

Find the area of XYZ.

A = 195 cm2

Find the area of parallelogram PQRS with vertices P(1, 2), Q(6, 2), R(8, 5), and S(3, 5).

The Pythagorean Theorem and Its Converse

Lesson 7-2

Pythagorean Thm

If a triangle is right, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

a2 + b2 = c2

b

a

c

GSP

How high up on the wall will a twenty-foot ladder reach if the foot of the ladder is placed five feet from the wall?

Pythagorean Triples Any set of three whole numbers that satisfy the

Pyth. Thm. are called a Pythagorean Triple. Which of the following are?

Using the Pythagorean Thm.

A right triangle has legs of length 16 and 30. Find the length of the hypotenuse. Do the lengths of the sides form a Pythagorean triple?

34. Yes.

summary

So, if a2 + b2 = c2 and a, b, & c are integers, then a, b, & c form a pythagorean triple

Properties of Exponents

Multiplication Multiplication Property of Property of ExponentsExponents

Power Power Properties of Properties of ExponentsExponents

Division Division Property of Property of ExponentsExponents

bm·bn = bm+n(bm)n = bmn

(ab)n = anbn

mnm

nb

b

b

Express each square root in its simplest form by factoring out a perfect square.

12 18 24 32 40

48 60 75 83 85

Express each product in its simplest form.

223

234

232

3263237

More practice simplifying expressions

1. 2.

3. 4.

369 3250

7218 238

example

Find the value of x. Leave your answer in simplest radical form.

112x

Example 4: SAT

In figure shown, what is the length of RS?

7

3

RT

S

Finding Area

The hypotenuse of an isosceles right triangle has length 20 cm. Find the area.

1002102102

12

1

210

A

bhA

x

Real World Connection

A baseball diamond is a square with 90-ft sides. Home plate and second base are at opposite vertices of the square. About how far is home plate from second base?

Stinks!!!Boooo…

Converse of the Pythagorean Thm.

If the square of the length of one side of a triangle is equal to the sum of the lengths of the other two sides, then it is a right triangle.

GSP

Example

Which of the following is a right triangle?

Acute Triangle TheoremAcute Triangle Theorem

If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then it is an acute triangle.

Obtuse Triangle TheoremObtuse Triangle Theorem

If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then it is an obtuse triangle.

Classifying

The numbers represent the lengths of the sides of a triangle. Classify each triangle as acute, obtuse, or right.

a. 15, 20, 25 b. 10, 15, 20

Right Obtuse

Example 5

Can segments with lengths 4.3 feet, 5.2 feet, and 6.1 feet form a triangle? If so, would the triangle be acute, right, or obtuse?

Assignment

Pg. 3602-44 even, 48-51, 76-77

Classwork/Homework

Pg. 3511,3, 9-23 odd, 26, 30-32, 44-46, 49

Pg. 3601-43 odd, 44, 48-53, 76-77