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Transcript of Pythagorean Theorem!
Theorem ! Diagrams : SOLUTION : By: Raveena
809 Important Words : Pythagoras' Law : How Pythagorean Theorem Works : History About the
Pythagorean Theorem: Real World
Application(s) : QUESTION : The pythagorean theorem (py·thag·o·re·an the·o·rem) :
A theorem attributed to Pythagoras that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other (ONLY WORKS ON RIGHT ANGLE TRIANGLES) Hypotenuse (hy·pot·e·nuse) :
The longest side of a right triangle that is placed directly opposite of the right angle (MUST ALWAYS BE LABELED ''C'') Right Angle Triangle (right an-gle tri·an·gle) :
A triangle with a right angle (90°) The legs of a triangle (legs of a tri·an·gle) :
The two shorter sides of a right angle triangle; perpendicular to one another (LABELED EITHER ''A'' OR ''B'') a + b = c
c - b = a
c - a = b It is a proven
fact that if you have at least 2 of the numbers (doesn't matter which: a ,
b or c ), it is certain that you will be able to use this theorem and figure out the 3 and final number! 2 2 2 2 2 2 2 2 2 2 2 2 The pythagorean theorem was
created by, and named after, Greek mathematician: Pythagoras, in 530 B.C. ; where he discovered the relationship between the 3 sides of a right angle triangle! Why The Pythagorean Theorem is IMPORTANT : The pythagorean theorem can be used to find the 3 side of a triangle if you know the measures of both of the other 2 sides of the triangle; proving that both of the legs of a triangle squared are equal to the hypotenuse rd squared. The pythagorean Theorem is important because ... Firefighter Construction worker For example ( If needed to climb into apartment window ):
You would need to know how high the building is and how tall the ladder is. You would need to figure out how far away the ladder needs to be placed. For example ( If building a roof ) :
You would need to know the width of the house's
dimension and the length of the house's dimension, in order to figure out how long the shingles need to be and how many of them (noting that it needs to be half of the house's width). The are just some of the jobs that use the pythagorean theorem in their daily jobs! Other jobs : Truck drivers, Painters, etc Raveena is decorating a ballroom ceiling with garland. The width of the rectangular ceiling is 8 metres, and the diagonal distance from one corner to the opposite corner is 10 metres. How much garland will Raveena need for the length of the ceiling? 1) Draw a diagram a = ? b = 8 c = 10 m m m 2) Write out the appropriate formula a = c - b 2 2 2 3) Fill in the blanks that we already know a = 10 - 8 2 m m 2 2 4) Solve a = 10 - 8
a = 10 x 10 - 8 x 8
a = 100 - 64
a = 36
a = √36
a = 6 m m m m m m m m m m m 2 2 2 2 2 2 Raveena will need 6 metres of garland. Angles ! QUESTION : ANGLES Key Geometry
Terms Adjacent Angles -> two angles that share a common vertex and side, but have no common interior points Supplementary Angles -> Two angles whose sum is 180 degrees Complimentary Angles -> Two angles whose sum is 90 degrees Opposite Angles -> the angles opposite to one other (are formed when
two lines intersect); equal to one another Transversal -> A line that cuts across two or more (usually parallel) lines Straight Angles -> an angle that is 180° exactly Perpendicular Lines -> Two lines that intersect to form right angles Corresponding angles -> two nonadjacent angles made by the crossing of two lines by a third line, one angle being interior, the other exterior, and both being on the same side of the third line. Right angle -> an angle that measures exactly 90 degrees Obtuse Angle -> an angle that is greater than 90° but less than 180° Acute Angle -> an angle that is less than 90° Reflex Angle -> an angle that is greater than 180° Interior of an angle -> The set of all points between the sides of an angle. Exterior of an angle -> The set of all
points outside an angle. To get from point A to point B Raveena must avoid walking through a pond. To avoid the pond, Raveena must walk 34 meters south and 41 meters east. To the nearest meter, how many meters would be saved if she walked through the pond? QUESTION : SOLUTION : 1) Draw a diagram 2) Write out the appropriate formula a + b = c 2 2 2 3) Fill in the blanks that we already know 34 = 41 = c 2 2 2 m m m 4) Solve m m m m m m m m m m 2 2 2 2 2 2 c = 34 + 41
c = 34 x 34 + 41 x 41
c = 1156 + 1681
c = 2837
c = √2837
c = 53 34m + 41m = 75m
75m - 53m = 22m CONCLUSION : In order to get from point A to point B (through the pond), Raveena will walk 53 meters. Because the distance from walking all of the way around is a total of 75 meters, you would subtract 53 meters; Raveena would have saved 22 meters if she walked through the pond. Angles are measured in degrees (written °). The maximum angle is 360°. This is the angle all the way round a point. Half of this is the angle on a straight line, which is 180°; there are many more angles but these 2 are the most common used. Angles : GENERAL Angles : HISTORY Angles have a long and diverse history. Angles were measured in degrees and were based around the number 60. This base was also used to measure time
(60 seconds = 1 minute, etc.). ABC is a right angle. What is the value of x?
SOLUTION : Since ABC is a right angle, the two angles must be complementary i.e. they must add to 90°
Therefore, the 2 angles are 11° and 79°
So (2x - 33)° + (5x - 31)° = 90°
7x - 64 = 90
7x = 90 + 64 = 154
x = 154 ÷ 7 = 22