Special RightTriangles

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Special RightTriangles
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There are 2 special right triangles

• The first one well talk about is a
• 45-45-90

The length of the hypotenuse will always be
bigger by v2
BEHOLD an isosceles triangle
xv2
x
The length of the sides are congruent lets call
them x.
x
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Lets try
mAC 12 mCB ____ mAB ____
12
12v2
5v2
12v2
12
5v6
2
2v2
2
2v2
mAC 2 mCB ____ mAB ____
5
5v3
12
2
5
5v3
mAC ____ mCB ____ mAB 5v2
5
5
5v3
5v3
mAC ____ mCB ____ mAB 5v6
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Given the leg, multiply it by v2 to find the
hypotenuse.

• Given the hypotenuse, divide it by v2 to find the
  leg.

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The 2nd special right triangle is a 30-60-90
Shortness rules!
Size matters.
The lengths of the sides of this triangle are
based on the SHORT SIDE.
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Fun fact The smallest angle is across the
shortest side.
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LETS TRY THIS

• X 6 Y _____ Z _____

6 v3
12
Z
60
X
2 v3
X _______ Y 6 Z ________
4 v3
30
Short side first!!
Y
4 v2
8
P ______ Q 4v2 R _____
5
5v2
P 5 Q ________ R ________
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LETS TRY THIS
4 v2
8
P ______ Q 4v2 R _____
5
5v2
P 5 Q ________ R ________
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There are 2 special right triangles 45-45-90
30-60-90
In a 30-60-90 triangle, the length of the
hypotenuse is 2 times the length of the shorter
leg AND the length of the longer leg is v3 times
the length of the shorter leg.

• In a 45-45-90 triangle, the length of the
  hypotenuse is v2 times the length of the leg.

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Find the area game?

• A ½ bh What is the length of the missing
  side?

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A ½ (6) (6) A 18 sq. units
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A ½ bh Find the short side first!!
A ½ (5) (5v3) Find an exact answer A 25v3 sq.
units or 12.5v3 2
5v3
5
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Areas of Regular Polygons
What part of A1/2bh is the perpendicular
bisector?
Can you find the area of a triangle?

• The perpendicular bisector of a triangle in a
  polygon is called an APOTHEM.
• The formula for the area of a regular polygon is
  
• A ½ap
• a is the length of the apothem
• p is the perimeter of the polygon

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Lets see how this works
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• A 1/2ap
• A ½(6.88)(50)
• A 172 sq.units

6.88
PAINLESS!! Lets kick it up a notch
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Find the area of this one!
Hmmmmm.. A circle has 360
Hmmmmm.. How many degrees would the top angle of
each ? have?
Hmmmmm.. Since the ?s are isosceles, what are
the measures of the base angles?
60
30
Hmmmmm.. If the apothem is an angle bisector,
then what is the measure of the small top angle?
The short side 6 The apothem 6v3 A 1/2ap
A ½(6v3)(72) 216v3 (exact) A 374.12
(approx.)
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WHEW!Try this

• Find the perimeter and area of a 30-60-90 ? with
  a hypotenuse of 18units.
• (sketch it)

What is the length of the short side? What is the
length of the long leg? What is the perimeter?
(exact) 27 9v3 units What is the area?
(exact) 81v3 sq units or 40.5v3 units2
2
60
9
30
9v3
(hint the smallest angle is across from the
smallest side the largest angle is across from
the largest side.)
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Whiteboard time!!
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Assignmentpg 336, 10-21