Algebra Period 3 (Wednesday)

ADDING AND SUBTRACTING RADICALS 11-6:

Again, radicals function like variables, so you can only COMBINE LIKE RADICALS!
You cannot add SQRT 2 to SQRT 3!!!!

However, you may add (3 SQRT 2) to (5 SQRT 2) and get 8 SQRT 2:
(3 + 5) SQRT 2 = 8 SQRT 2

Make sure you simplify all radical expressions before trying to combine them!
Sometimes, it looks like they are not like radicands, but then after simplifying they are.

EXAMPLE #34 from p. 505:
SQRT( x²y) + SQRT( 4x²y) + SQRT(9y) - SQRT( y³)

Simplify each term first!!!!!!!!

x SQRT( y) + 2x SQRT( y) + 3 SQRT( y) - y SQRT( y)

NOW THEY ARE ALL LIKE TERMS BECAUSE ALL HAVE SQRT y
(x + 2x + 3 - y) SQRT y
(3x - y + 3) SQRT y (final simplified answer)

PYTHAGOREAN THEOREM 11-7 (an old friend)

FOR RIGHT TRIANGLES ONLY!

2 legs - make the right angle - called a and b
(doesn't matter which is which because you will add them and adding is COMMUTATIVE!)
hypotenuse - longest side across from the right angle - called c
You can find the third side of a right triangle as long as you know the other two sides:
a² + b² = c²
After squaring the two sides that you know, you'll need to find the square root of that number to find the length of the missing side (that's why it's in this chapter!)

EASIEST - FIND THE HYPOTENUSE (c)
Example #1 from p. 510
8² + 15² = c²
64 + 225 = c²
289 = c²
c = 17

A LITTLE HARDER - FIND A MISSING LEG (Either a or b)
Example #5 from p. 510
5² + b² = 13²
25 + b² = 169
b² = 169 - 25
b² = 144
b = 12

DISTANCE FORMULA
(based on the Pythagorean Theorem): see p. 513 in book

The distance between any two points on the coordinate plane (x y plane)
The distance is the hypotenuse of a right triangle that you can draw using any two points on the coordinate plane (I'll show you how to draw it in class).
The formula is:
distance = SQRT[( difference of the two x's)² + (difference of the two y's)²]
The difference between the 2x’s is the length of the leg parallel to the y axis and
the difference between the 2y’s is the length of the leg parallel to the x axis

EXAMPLE: What is the distance between (3, -10) and (-7, -2)?
d = SQRT[(3 - -7)² + (-10 - -2)²]
d = SQRT[10² +( -8²)]
d = SQRT(164)
Simplifying: 2SQRT41

USING THE PYTHAGOREAN THEOREM - WORD PROBLEMS 11-8
There are many real life examples where you can use the Pythagorean Theorem to find a length.

EXAMPLE: HOW HIGH A 10 FOOT LADDER REACHES ON A HOUSE?

A 10 ft ladder is placed on a house 5 ft away from the base of the house.
Find how high up the house the ladder reaches.
The ladder makes a right triangle with the ground being one leg, the house the other, and the ladder is the hypotenuse ( see drawing in #1 on p. 515) You need to find the distance on the house, so you're finding one leg.

FLYING A KITE? HOW LONG MUST THE STRING BE?

You're flying your kite for the kite project and you want to know how long the kite string must be so that it can reach a height of 13 ft in the air if you're standing 9 feet away from where the kite is in the air. The string represents the hypotenuse. You know one leg is the height in the air (13 ft) and the other leg is how far on the ground you are standing away from where the kite is flying (9 ft)
You need to find the hypotenuse.