Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right...

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Right Triangles Right Triangles And the And the Pythagorean Pythagorean Theorem Theorem

Transcript of Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right...

Page 1: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

Right TrianglesRight Triangles

And the And the

Pythagorean TheoremPythagorean Theorem

Page 2: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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

Page 3: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle 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.

Page 4: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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.

Page 5: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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

Page 6: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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.

Page 7: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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

Page 8: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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.

Page 9: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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

Page 10: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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.

Page 11: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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

Page 12: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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.

Page 13: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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

Page 14: Right Triangles And the Pythagorean Theorem. Legs of a Right Triangle Leg -the two sides of a right triangle that form the right angle Leg.

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.