Geometry Unit 2: Area, Perimeter, Pythagorean Theorem, Square...

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Geometry Pre-Unit 1 Intro: Area, Perimeter, Pythagorean Theorem, Square Roots, & Quadratics

P-1 Square Roots and SRF

Square number – the product of a number multiplied by itself.

1 * 1 = 1 1 is a square number



List the first twelve square numbers:

____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____

Square Root – a number that when multiplied by itself is a square number.

4 is the square root of 16




What are the square roots of the following numbers?

A) 225 B) 49 C) 81 D) 121 E) 144 F) 576 G) 900

The following symbol (radical) is used for square roots:

When you see that symbol it means find the square root of the number.

Examples:

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Simplest Radical form – a radical that is not a perfect square but has been reduced to remove all

squares from the radicand.

Simplest radical form is a way of breaking up a radicand into primes then square rooting.

For each pair of primes in the final breakdown, place one digit in front of the radical as a product and the

radicand in the root is multiplied back together in the final result

Examples

Multiplying Radicals - Multiply what is outside the radical together and multiply what is under the radical

together.

Examples

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P-2 Adding and Subtracting “like roots”

Like roots – roots with the same radicand

x and x can be added together to make 2x, 3y and 3y can be added together to make 6y and in the same way

like roots can added together.

Examples:

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P-3 Pythagorean Theorem

Pythagorean Theorem- The sides of a right triangle are related according by the equation: a2 + b2 = c2

This is significant because you can find the third side of a right triangle if given the other two sides.

a and b are the short sides of the triangle called legs

c is the long side called the hypotenuse

*c is the side not adjacent to the right angle

Example 1: Find the length of the hypotenuse of the right triangle given both legs.

Leg = 8in Leg = 6in

Example 2: Find the missing leg of the right triangle given the length of the hypotenuse and a leg.

Hypotenuse = 17 Leg = 13in

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Example 3: Find the length of the hypotenuse given the legs are 6 in.

Step 1: Find the hypotenuse using the Pythagorean Theorem

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P-4 Midpoint and Distance Formulas

Midpoint formula:

Distance Formula:

Find the midpoint of A(-1, 7) and B(4, 13) is calculated using the midpoint formula:

The distance between of A(-1, 7) and B(4, 13) is calculated using the distance formula:

Class Activity: The following is a mathematical proof

Given points: A(2, 6) and B(18, 14)

Find the midpoint C and prove that it bisects .

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P-5 Perimeter

Perimeter – the sum of the lengths of the sides of a polygon (shape)

The perimeter of the square at right would be:

1.5in + 1.5in +1.5in + 1.5in = 6in

The formula for perimeter of a square is P = 4s

P stands for perimeter and S stands for side

P = 4(1.5) = 6in

What are the perimeters of the following squares?

A) 3ft on a side B) seven halves inches on a side C) 2.5 yd on a side D) 2 m on a side

The formula for perimeter of a rectangle is P = 2b + 2h

h stands for height and b stands for base

The perimeter of the rectangle is

P = 2(3.2ft) + 2(2.5ft) = 6.4 + 5 = 11.4ft

What would be the perimeter of the following rectangles? Draw a picture first

A) base = 3 in B) base = 2 ft C) base = 5.6m

height = 8 in height = 3.7 ft height = 3m

Extension: The base of the Great Pyramid in Egypt is a square measuring 756ft per side.

What is the perimeter of the base of the pyramid?

1.5in

b

h

3.2 ft

2.5 ft

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Equilateral Triangle – A triangle with all side lengths congruent.

What is the perimeter of the figure at right?

P = 3(0.5) = 1.5ft 0.5ft

What would be the perimeter of an equilateral triangle with:

A) side length w B) side length C) side length 14km

Isosceles Triangle – A triangle with two congruent sides.

The triangle at right is isosceles.

The congruent sides are called legs.

The non-congruent side is called the base.

What would the perimeter be of an isosceles triangle with the following measurements?

A) legs – 4in, base – 5in B) legs – 1.5ft, base – 12ft C) legs – 2ft, base – 5.4ft

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P-6 Circumference & Area

Circumference – the perimeter of a circle

Area – the number of square units that it would take to fill an object.

Formula’s: C = 2πr or C = dπ A = πr2

π is about 3.14

Radius – half the length of the diameter (r) 2r = d

Diameter – the length across a circle through the center (d)

The diameter is 4ft The radius is 1ft

C = dπ C = 2πr A = πr2

C = 4π ≈ 12.56ft C = 2π ≈ 6.28ft A = π ≈ 3.14 ft2

Try these: Find the circumference and area of the circles below given the information:

A) radius = 3m B) diameter = 7mi C) radius = 2in

Compound Shapes – A shape that contains more than one polygon.

The perimeter of this shape is a combination of three sides of a rectangle and a semicircle. Use 3.14 for π

Perimeter of the three sides: Perimeter of the semicircle:

Sum of the perimeters:_________________

A 1ft

4ft

A

1.3 ft

2 ft

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P-7 Area of Polygons

Area – the number of square units that it would take to fill an object.

Rectangle: A = bh Square: A = s2

A = (3)(14) = 42km2

height = 3km A = 1.52 = 2.25cm2

The area of the rectangle is 42km2. The area of the square is 1.5cm2.

A) Rectangle: b = 4in, h = 2.5in B) Square s = 1.2mi C) Rectangle: b = 2ft, h = 3.4ft

Triangle: A = 0.5 bh

h h h

b

b b

Acute triangle Right triangle Obtuse triangle

All angles less than 90 One angle exactly 90 One angle larger than 90

Height is inside Height is the edge of the right angle Height is outside the triangle

base = 14km

height = 3km

3ikm

side = 1.5cm

3ikm

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Examples:

3cm 3.1in 1.1ft

4cm 5in 2.8ft

Parallelogram: A = bh

b

h h

b

NOTE: The base and height must meet at a perpendicular.

3 in

4ft 4in

5.1ft

Trapezoid: A = 0.5h(b1+b2) It does not matter which base you label as b1 or b2

h

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S A

H

W

T

Z

L

J K

M

2.5 ft

8ft

A =

6 ft

32 ft

34ft

A =

48 ft

Rhombus: A= 0.5(d1d2) d=diagonal

Kite A= 0.5(d1d2)

Solve for the Area of the rhombus if A) JL = 32cm and MK = 26cm B) JZ = 7, LZ = 7, MZ = 5, and KZ = 5

Solve for the Area of the kite if A) WA = 12ft and HT = 34ft B) HS = 4, TS = 16, WS = 5 and AS = 5ft

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If each side of the pentagon at right is congruent and

the length of one side is 6m. The apothem is 3.4m.

What is the Area of the regular polygon?

Regular Polygon: A = 0.5 aP

P = Perimeter a = apothem

\

a) Find the Area of a regular octagon with the length one side is 10 cm and the apothem is 12.1 cm?

b) Find the Area of a regular heptagon with the length one side is 4 cm and the apothem is 3.1 cm?

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P-8 Using the Quadratic Formula

Quadratic Formula

If 2 0, 0ax bx c a , and

2 4 0,b ac then

2 4

2

b b acx

a

Solve by factoring: Use the quadratic formula to solve:

2 6 8 0x x 22 7 15 0x x

23 8 0n n 23 3 0x x

Solve the quadratic by factoring, by taking the square root, completing the square, or the quadratic formula.

2 4 4 0n n

Geometry Unit 2: Area, Perimeter, Pythagorean Theorem, Square...

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