Download Apply the Pythagorean Theorem

Apply the Pythagorean Theorem
Chapter 7.1
Sides of a Right Triangle
• Hypotenuse – the side of a right triangle
opposite the right angle and the longest side.
• Legs – the sides of a right triangle that are not
the hypotenuse.
The Pythagorean Theorem
• In a right triangle, the square of the length of
the hypotenuse is equal to the sum of the
squares of the legs.
Proving the Pythagorean Theorem
• There are multiple ways of proving the
Pythagorean Theorem.
• If you can find a legitimate proof of the
Pythagorean theorem you may earn 2 bonus
points.
Using the Pythagorean Theorem
6a  8b  cx²
2
2
2
36 + 64 = x²
100= x²
100  x
2
10  x
Find the missing leg
xa  3b  c5²
2
2
2
x²+ 9 = 25
x² = 16
x  16
2
x4
Find the Area of the Triangle
What is the formula for the area of
a triangle?
A = ½bh
How will we find the height?
5  h  13
2
25  h  169
2
h  144 h  12
1
1
A  bh  (10)(12)  60
2
2
2
2
2
Find the Area of the Triangle
4  h  12
2
16  h  144
2
2
2
h  128
2
h 8 2
1
1
A  bh (8)(8 2 ) 32 2
2
2
Pythagorean Triples
• A Pythagorean Triple is a set of 3 positive
integers or whole numbers that satisfies the
Pythagorean theorem.
Is it a Pythagorean Triple?
• 3, 4, and 5 32  4 2  52
9  16  25
25  25
• 21, 28, and 35 yes
• 30, 72, and 91 no
• 14, 48, and 50 yes
yes
If I am given 2 sides of a right triangle
make up the sides of Pythagorean triple,
how do you find the missing side?
There are 2 possible scenarios:
1. You are given both legs of the right triangle
and need to solve for the hypotenuse.
2. You are given one leg and one hypotenuse
and need to solve for the other leg.
Given 2 sides: 20 and 25
•
•
•
•
•
•
•
Scenario 1.
20 and 25 are the legs
202 + 252 = x2
400 + 625 = x2
1025 = x2
32.02 = x
This cannot be the
answer for a
Pythagorean triple
because it is not a
whole number.
• Scenario 2.
• 20 and 25 are the leg
and hypotenuse
• 202 + x2 = 252
• 400 + x2 = 625
• x2 = 225
• X = 25
• This can be the answer
for a Pythagorean
triple because it is not
a whole number.
Given 2 sides: 28 and 96
Find the area when given a leg and the
hypotenuse
1. Find the other leg by plugging the known
values into the Pythagorean Theorem.
2. Use the 2 legs in the formula for area of a
triangle
 A = ½BH
Find the area when given a leg and a
hypotenuse.
•
•
•
•
L=8 and h= 16
Aò+8ò=16ò
Aò+64=256
Aò=192
a  192
2
1
1
bh   8  192  55.42
2
2
Find the area when given a leg and a
hypotenuse.
•
•
•
•
L=13 and h= 17
Aò+13ò=17ò
Aò+169=289
Aò=120
a  120
2
1
1
bh  13  120  71.2
2
2
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