The Pythagorean Theorem: The Pythagorean Theorem A tool for right triangle problems only
When do I need to use the Pythagorean Theorem ?: When do I need to use the Pythagorean Theorem ? When I know: length of 2 sides And Need to know: length of 3rd side
PowerPoint Presentation: What is the Pythagorean Theorem? a b c Where a and b are legs and c is the hypotenuse
Using a2 + b2 = c2: Using a 2 + b 2 = c 2 Looking for length of the hypotenuse 15 20 x a 2 + b 2 = c 2 15 2 + 20 2 = c 2 225 + 400 = c 2 625 = c 2
Using a2 + b2 = c2: Using a 2 + b 2 = c 2 Looking for length of A leg 6 x 10 a 2 + b 2 = c 2 6 2 + b 2 = 10 2 36 + b 2 = 100 -36 -36 b 2 = 64 b = 8
Applying the Pythagorean Theorem: Applying the Pythagorean Theorem
A 15 foot ladder leans up against a building. The foot of the ladder is 5 feet from the base of the building. How high up the wall, to the nearest foot does the ladder reach?: A 15 foot ladder leans up against a building. The foot of the ladder is 5 feet from the base of the building. How high up the wall, to the nearest foot does the ladder reach? Draw a picture: 15 5 x
Solving the problem: Solving the problem a 2 + b 2 = c 2 Write formula x 2 + 5 2 = 15 2 Substitute in x 2 + 25 = 225 Solve for x - 25 -25 x 2 = 200 x = 14.142135 Check: How am I to leave my answer? The ladder reaches 14 feet up the wall.
Tim rode 8 miles due north, then 3 miles due east. How far, to the nearest mile, is Tim from where he started? : Tim rode 8 miles due north, then 3 miles due east. How far, to the nearest mile, is Tim from where he started? Draw a picture: 8 3 x
Solve the problem: Solve the problem a 2 + b 2 = c 2 8 2 + 3 2 = c 2 64 + 9 = c 2 73 = c 2 Remember: How am I suppose to leave my answer? C = 8.5440037 Tim is 9 miles from where he started.
Further Applications: Further Applications The diagonals of a rhombus are 6 cm and 8 cm. What is the length of each side of the rhombus? Prior Knowledge: What properties does a rhombus have? The diagonals of a rhombus: bisect each other The diagonals of a rhombus: are perpendicular The sides of a rhombus: are congruent
Draw a picture and solve:: Draw a picture and solve: a 2 + b 2 = c 2 3 2 + 4 2 = c 2 9 + 16 = c 2 25 = c 2 3 3 4 4 x 5 = c Each side of the rhombus is worth 5 cm. A rhombus is really 4 right triangles in disguise!
As seen in the accompanying diagram, a person can travel from NYC to Buffalo by going north 170 miles to Albany and then west 280 miles to Buffalo. A) If a highway is built to connect NYC and Buffalo, how many miles would be saved on the trip? B) With gas prices at $3.10 and a vehicle that gets 18 mpg, how much money would be saved roundtrip, if the new highway was traveled instead of the old route?: As seen in the accompanying diagram, a person can travel from NYC to Buffalo by going north 170 miles to Albany and then west 280 miles to Buffalo . A) If a highway is built to connect NYC and Buffalo, how many miles would be saved on the trip? B) With gas prices at $3.10 and a vehicle that gets 18 mpg, how much money would be saved roundtrip, if the new highway was traveled instead of the old route?
Find length of new highway: Find length of new highway Buf Albany New York City 170 miles 280 miles ??? a 2 + b 2 = c 2 280 2 +170 2 =c 2 107300 m= c 2 327.566= c Did I answer question? How many miles would be saved? Old Distance: 280 + 170 = 450 New Distance: 327.566 Saved Miles: 122.4 or 122 miles
How much money can be saved?: How much money can be saved? Saved Miles: 122 miles x 2 = 244 Cost to drive one mile (gas): $3.10 divided by 18. ($0.1722…) Cost to drive 244miles $ 0.1722 times 244 Saved: $42.02
Wrapping It UP: Wrapping It UP The Pythagorean Theorem can be used only on _____triangles. When should the Pythagorean Theorem be used? What should be done first when solving a word problem involving the Pythagorean Theorem? What must be done before writing the answer to a Pythagorean Theorem problem? Right When the length of 2 sides are known and the length of 3rd side is needed Draw and label triangle Check to see whether the answer should be rounded or not