right triangles and the pythagorean theorem. legs of a right triangle leg -the two sides of a right...
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Right TrianglesRight Triangles
And the And the
Pythagorean TheoremPythagorean Theorem
Legs of a Right TriangleLegs of a Right Triangle
Leg -the two sides Leg -the two sides of a right triangle of a right triangle that form the right that form the right angleangleL
eg
Leg
HypotenuseHypotenuse
In a right In a right triangle, the side triangle, the side opposite the right opposite the right angle is the angle is the hypotenuse.hypotenuse.
TriangleTriangle
The sum of all The sum of all angles measure angles measure 180180° in a ° in a triangle.triangle.
A right triangle A right triangle has a 90° angle.has a 90° angle.
Pythagorean TheoremPythagorean Theorem
The Pythagorean Theorem The Pythagorean Theorem is used to find the measure is used to find the measure of an unknown side of a right of an unknown side of a right triangle.triangle.
Legs are side a and side b.Legs are side a and side b.
Hypotenuse is side c.Hypotenuse is side c.
The Pythagorean Theorem: The Pythagorean Theorem:
aa² + b² = c²² + b² = c²
Pythagorean TheoremPythagorean TheoremThe legs of a right triangle The legs of a right triangle
measure 6 in. and 8 in. measure 6 in. and 8 in. What does the What does the hypotenusehypotenuse measure? measure?
aa² + b² = c²² + b² = c² 6² + 8² = c²6² + 8² = c² 36 + 64 = c²36 + 64 = c² 100 = c²100 = c² √ √100 = √c²100 = √c² 10 = c10 = cThe hypotenuse is 10 in.The hypotenuse is 10 in.
Practice ProblemPractice Problemaa² + b² = c²² + b² = c²
3² + 4² = c²3² + 4² = c²
9 + 16 = c²9 + 16 = c²
25 = c²25 = c²
√ √25 = √c²25 = √c²
5 = c5 = c
3 cm
4 cm
C
Pythagorean TheoremPythagorean Theorem
aa² + b² = c²² + b² = c²
9² + b² = 15²9² + b² = 15²
81 + b² = 22581 + b² = 225
-81-81 -81-81
b² = 144b² = 144
√ √b² = √144b² = √144
b = 12b = 12
To find the measurement of a leg, we use the Pythagorean Theorem with leg a or b and the hypotenuse.
Practice ProblemsPractice Problems
aa² + b² = c²² + b² = c²
12² + b² = 13²12² + b² = 13²
144 + b² = 169144 + b² = 169
-144-144 -144-144
b² = 25b² = 25
√ √b² = √25b² = √25
b = 5b = 5
12 mm13 mm
b
18 cm
30 cma
a² + b² = c²a² + b² = c² a² + 18²a² + 18² = 30²= 30² a² + 324 = 900a² + 324 = 900 -324-324 -324-324 a² = 576a² = 576 √ √a² = √576a² = √576 a = 24a = 24
Pythagorean Theorem ReviewPythagorean Theorem Review
aa² + b² = c²² + b² = c²
Leg a and leg b Leg a and leg b represent the sides represent the sides of the right triangle.of the right triangle.
Side c is the Side c is the hypotenuse of a hypotenuse of a right triangle.right triangle.
Finding the HypotenuseFinding the Hypotenuse
aa² + b² = c² ² + b² = c²
7² + 10² = c² 7² + 10² = c²
49 + 100 = c² 49 + 100 = c²
149 = c²149 = c²
√ √149 = √c² 149 = √c²
12.2 ≈ c12.2 ≈ c
Pythagorean TheoremPythagorean Theorem
What is the diagonal length of a TV screen What is the diagonal length of a TV screen whose dimensions are 80 x 60 cm?whose dimensions are 80 x 60 cm?
aa² + b² = c²² + b² = c²
80² + 60² = c² 80² + 60² = c²
6,400 + 3,600 = c²6,400 + 3,600 = c²
10,000 = c²10,000 = c²
√ √10,000 = √c²10,000 = √c²
100 = c100 = c
The diagonal length of the TV screen is 100 cm.The diagonal length of the TV screen is 100 cm.
Finding the Leg of a Right TriangleFinding the Leg of a Right Triangle
aa² + b² = c²² + b² = c²
11² + b² = 18²11² + b² = 18²
121 + b² = 324121 + b² = 324
-121-121 -121-121
b² = 203b² = 203
√ √b² =√203b² =√203
b ≈ 14.2b ≈ 14.2
Pythagorean TheoremPythagorean TheoremHow far up a wall will an 11m How far up a wall will an 11m
ladder reach, if the foot of the ladder reach, if the foot of the ladder must be 4m from the ladder must be 4m from the base of the wall?base of the wall?
aa² + b² = c²² + b² = c² 4² + b² = 11²4² + b² = 11² 16 + b² = 12116 + b² = 121 -16-16 -16-16 b² = 105b² = 105 √ √b² =√105b² =√105 b ≈ 10.2b ≈ 10.2
The ladder will reach 10.2 meters up The ladder will reach 10.2 meters up the wall.the wall.