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1
Vocabulary:
3 5
0.75 which can bewritten as aratio is called arational number
and 2 which can't bewritten as ratios arecalled irrational numbers
or or thesymbol is called aRADICAL
n thenumber beneath the
t
radical sign is theRADICAND
n is theprincipal or positivesquareroot of n
n is thenegativesquareroot of n
n isthepositiveor negativesquareroot of n
÷
± n is thepositiveor negativesquareroot of n ±
What is the approximate value of these radicals? Think about
how close they are to the

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2
Solving Equations with Radicals
2
there are actually two solutions. Can you think of them both?
x4
which is written as 2
x2 or 2
The rest of the steps in solving equations just use the reverse operations that we have used before.
2
2x39
2
add 3 on both 2x12 sides
2
divide 2 on both s6sx ide
square root on both sx6 ides
2
3x152
2
add 1 on both 3x6 2sides
2
multiply by 2 on both 3x12sside
divide by 3 on both s4sx ide
square root on both sdsxie
2
2
3x154
2
multiply by 4 on bot3x12h 0essid
2
add 1 on both 3x21 sides
2
divide by 3 on both s7sx ide
suare root on both sx7 ides
2
2x45117
2
minus 5 on bot2xh 46s7side
2
72x o44n.s2 b
2
4 o2x3.s8n b
2
2 onsx19 b.
x19
2
4.2x165
x7
2
3x2452.
x10
2
3(m1)63.2
m3
2
1.5y411
y2
2
4g21035.
g3
2
5(h1)16.9
h3
2
y.273
3
y2
22
5n11382n.
n2
2
3z289.5
z10
2
53b110.2
bno solution
2
3(5z1411.2)7
z2
2
52(4t11121.)
t1
Finding the Area of Complex Shapes–see Looking For Pythagoras. Investigation 2
1. Make a rectanglearound the shape.2. Make triangles and rectangles in the outside.
A
BCDE
n e area o e recane.
Find the area of the outside shapes.12B12
11A0.52
C212
22D22
31E1.52
Find the total outside area7
Subtract the area of the outside fromthe rectangle area to get the inside area.5
Find the area of the rectangle.5420
BC ADE
Find the area of the outside shapes.B212
32A32
12C12
41E22
Find the total outside area14
Subtract the area of the outside fromthe rectangle area to get the inside area.6
34D62
3
Finding the Length of Segments–see Looking For Pythagoras. Page20
Make a squarearound the segment.Counting the number of dots on each side makes it easy.
n e area o e ouse square.
Find the area of the outside triangles.
)
514(102
Find the area of the inside square.361026
Therefore the length of one sideof the square is26n e area o e ouse square.
Find the area of the outside triangles.
)
324(122
Find the area of the inside square.251213
Therefore the length of one sideof the square is13
Pythagorean Theorem. See page 482 in the textbook.In a right triangle the area of a square on the hypotenuse is equal to the sum of the area of the squares on the other two sides. Note: The hypotenuse is the long side of the triangle. It is opposite the right angle.
222
abc
The sum of the areas gives the equation:
When the hypotenuse is the unknown.
222
abc
222
35x
34x
x34
When a leg is the unknown.
222
abc
222
5x11
222
x115
2
x96
x96
x46
4
When the side lengths include variables.
222
abc
222
(2x)(3x)8
22
4x9x64
2x83x
2
13x64
64x13
813
222
abc
222
5p67p
222
25p649p
22
624p
2
p2
3 p2
When there is more than one triangle in the diagram.
222
98y
Choose the triangle where you know two sides. Label the third side with a variable.
2
145y
Note: We don’t need to know what y is. Solve the other triangle.
222
yx13
2
22
we know y145 so
145x13
22
x13145
2
x24
x24
x26
Dot Paper Problems:Find the perimeter of this shape.Construct right triangles and then solve using Pythagoras.
22
x
22
y5229
perimeter4132925
22
z422025
Find the perimeter for these shapes.
21062512
812520
333532
3105245
21321035282
621310
5132765751
Questions Involving Recognising Pythagorean Relationships:1. Recognising right triangles:Are the following triangles right triangles?The sides of a right triangle will make the Pythagoras equation true. Try it for this triangle
697
222
679
364981
8581
Not true so not a right triangle.
152016729054
222
151620
481400
Not true so nota right triangle.
222
121620
400400
True so it isa right triangle.
162012
222
547290
81008100
True so it isa right triangle.

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