Lesson #6: Triangles & Pythagorean Theorem Unit 3: Geometry Lesson #6: Triangles & Pythagorean Theorem

LEARNING GOALS Find missing sides in right triangles To determine if a triangle is right or not To explain Pythagorean theorem To explain how to and when to use the Pythagorean theorem

Classifying Triangles by Sides Scalene: A triangle in which all 3 sides are different lengths. BC = 5.16 cm B C A BC = 3.55 cm A B C AB = 3.47 cm AC = 3.47 cm AB = 3.02 cm AC = 3.15 cm Isosceles: A triangle in which at least 2 sides are equal. HI = 3.70 cm G H I Equilateral: A triangle in which all 3 sides are equal. GI = 3.70 cm GH = 3.70 cm

Classifying Triangles by Angles Acute: A triangle in which all 3 angles are less than 90˚. 57 ° 47 76 G H I Obtuse: 108 ° 44 28 B C A A triangle in which one and only one angle is greater than 90˚& less than 180˚

Classifying Triangles by Angles Right: A triangle in which one and only one angle is 90˚ Equiangular: A triangle in which all 3 angles are the same measure.

Classification by Sides with Flow Charts & Venn Diagrams polygons Polygon triangles Triangle scalene isosceles Scalene Isosceles equilateral Equilateral

Classification by Angles with Flow Charts & Venn Diagrams polygons Polygon triangles Triangle right acute equiangular Right Obtuse Acute obtuse Equiangular

Pythagoras Lived in southern Italy during the sixth century B.C. Considered the first true mathematician Used mathematics as a means to understand the natural world First to teach that the earth was a sphere that revolves around the sun

Right Triangles Longest side is the hypotenuse, side c (opposite the 90o angle) The other two sides are the legs, sides a and b Pythagoras developed a formula for finding the length of the sides of any right triangle

The Pythagorean Theorem “For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.” a2 + b2 = c2

Pythagoras’ Theorem This is the name of Pythagoras’ most famous discovery. It only works with right-angled triangles. The longest side, which is always opposite the right-angle, has a special name: hypotenuse

Pythagoras’ Theorem c a b c²=a²+b²

Pythagoras’ Theorem c c b a c²=a²+b² y a a b b a c c

Using Pythagoras’ Theorem What is the length of the slope?

Using Pythagoras’ Theorem c a= 1m b= 8m c²=a²+ b² ? c²=1²+ 8² c²=1 + 64 c²=65

Using Pythagoras’ Theorem c²=65 We need to use the square root button on the calculator. How do we find c? √ It looks like this √ Press , Enter 65 = So c= √65 = 8.1 m (1 d.p.)

Example 1 c b a 9cm c²=a²+ b² c²=12²+ 9² 12cm c²=144 + 81 c²= 225

Example 2 a b c 4m 6m c²=a²+ b² s²=4²+ 6² s s²=16 + 36 s²= 52 s = √52 =7.2m (1 d.p.)

Finding the shorter side 7m 5m h c c²=a²+ b² a 7²=a²+ 5² 49=a² + 25 ? b

Finding the shorter side 49 = a² + 25 + 25 We need to get a² on its own. Remember, change side, change sign! 49 - 25 = a² a²= 24 a = √24 = 4.9 m (1 d.p.)

Example 1 c b a c²= a²+ b² w 6m 13m 13²= a²+ 6² 169 = a² + 36 Change side, change sign! 169 = w² + 36 a 169 – 36 = a² a²= 133 a = √133 = 11.5m (1 d.p.)

Applications The Pythagorean theorem has far-reaching ramifications in other fields (such as the arts), as well as practical applications. The theorem is invaluable when computing distances between two points, such as in navigation and land surveying. Another important application is in the design of ramps. Ramp designs for handicap-accessible sites and for skateboard parks are very much in demand.

Baseball Problem A baseball “diamond” is really a square. You can use the Pythagorean theorem to find distances around a baseball diamond.

Baseball Problem The distance between consecutive bases is 90 feet. How far does a catcher have to throw the ball from home plate to second base?

Baseball Problem To use the Pythagorean theorem to solve for x, find the right angle. Which side is the hypotenuse? Which sides are the legs? Now use: a2 + b2 = c2

Baseball Problem Solution The hypotenuse is the distance from home to second, or side x in the picture. The legs are from home to first and from first to second. Solution: x2 = 902 + 902 = 16,200 x = 127.28 ft

Ladder Problem A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window?

Ladder Problem Solution First draw a diagram that shows the sides of the right triangle. Label the sides: Ladder is 25 m Distance from house is 7 m Use a2 + b2 = c2 to solve for the missing side. Distance from house: 7 meters

Ladder Problem Solution b = 24 m How did you do?

Success Criteria I can identify a right-angle triangle I can identify Pythagorean theorem I can identify when to use Pythagorean theorem I can use Pythagorean theorem to find the longest side I can use Pythagorean theorem to find the shortest side I can solve problems using Pythagorean theorem