The Pythagorean Theorem Section 8-1. Use the Pythagorean Theorem.
The Pythagorean Theorem
description
Transcript of The Pythagorean Theorem
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Triangle a b c a² b² c²
Blue* 5 12 13 25 144 169
Green 7 8 13 49 64 169
Orange* 3 4 5 9 16 25
Pink 6 5 6 36 25 36
Purple* 6 8 10 36 64 100
White* 7 24 25 49 576 625
Yellow* 8 15 17 64 225 289
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Pythagoras and his Theorem
• Right Triangle: a triangle with exactly one right angle.
• Legs: the sides of a triangle that form the right angle.
• Hypotenuse: the longest side, its located across from the right angle.
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c
a
b
The legs are labeled a & b and the
hypotenuse is ALWAYS labeled c.
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Pythagoras discovered that the sum of the squares of the two
legs in a right triangle is equal to the square of the hypotenuse.
That means, in any right triangle,
a² + b² = c² leg² + leg² =
hypotenuse²
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The term The term “squared” “squared” comes from the comes from the area of a area of a square.square.
EX:EX:
3 “squared” 3 “squared” means 3x3 or means 3x3 or 9.9.
The area of a The area of a 3x33x3 square is 9 square is 9
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Could a right triangle have sides that measure 3 cm, 4 cm, and 5
cm?a² + b² = c²3² + 4² ? 5² 9 + 16 ? 25 25 = 25
Yes, this is a right triangle because the Pythagorean Theorem works!
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How about sides of 5, 6, and 7?
a² + b² = c²5² + 6² ? 7² 25 + 36 ? 49
61 ≠ 49
NO, this is a NOT right triangle because the Pythagorean Theorem
doesn’t work!
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Is 15, 17, 8 a right triangle? Why or why not? Show Work!
a² + b² = c²8² + 15² ? 17² 64 + 225 ? 289
289 = 289
Yes, this is a right triangle because the Pythagorean Theorem works!
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Using the Pythagorean Theorem to Find a Missing Side
Note: the missing side is the hypotenuse
a² + b² = c²
5² + 12² = c²
25 + 144 = c²
169 = c²
√169 = √c²
13 = c
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What if you know the hypotenuse? You can use the theorem to find one of the legs.
a² + b² = c²9² + b² = 15² 81 + b² = 225-81 -81
b² = 144 √b² = √144 b = 12
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When your answers don’t work out evenly, round to the
nearest TENTHTENTH . a² + b² = c² a² + 4² = 11² a² + 16 = 121
-16 -16 b² = 105
√b² = √105 b = 10.246
10.2m
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Finally, we can use the Pythagorean Theorem to solve
real life word problems.
Jen hiked 8 miles east, then turned and hiked 6 miles south. How far was she from her starting
point?
DRAW A PICTURE!
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Jen hiked 8 miles east, then turned and hiked 6 miles south. How far was she
from her starting point?
8 miles east
6 miles
south
?
a² + b² = c² 8² + 6² = c² 64 + 36 = c² 100 = c² √100 = √c² 10 = c
Jen was 10 miles from where she started.
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Polygons
• Polygon: a closed figure formed by 3 or more line segments that intersect only at their verticies.
• Polygons are classified by the number of sides and angles they have
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Polygons
3 sides:3 sides:
triangletriangle4 sides:4 sides:
quadrilateral quadrilateral 6 sides:6 sides:
hexagonhexagon5 sides:5 sides:
pentagonpentagon
7 sides:7 sides:
heptagonheptagon10 sides:10 sides:
decagondecagon9 sides:9 sides:
nonagonnonagon 8 sides:8 sides:
octagonoctagon
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Regular Polygons • Regular Polygon: a polygon in which all the
sides are the same length and all the angles are the same measure.
Example:
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Interior Angles
Sum of Interior Angle Formula:Sum of Interior Angle Formula:
(n – 2) * 180(n – 2) * 180
What if we wanted to know the measure of EACH interior angle of a regular pentagon?
How would we go about doing this?Discuss with your partner.
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Find the measure of each interior angle of a pentagon.
108º108º
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Find the measure of each interior angle of a hexagon.
120º120º
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How many sides does a polygon have if the sum of its
interior angles is 1440º.
10 sides10 sides
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Find the measure of the missing angle in the figure below.
93º93º