The Pythagorean The Pythagorean TheoremTheorem

A A tooltool for for right right

triangletriangle problems problems onlyonly

When do I need to useWhen do I need to usethe Pythagorean the Pythagorean

Theorem ?Theorem ?When I know: length of 2 sidesWhen I know: length of 2 sides

AndAnd

Need to know: length of 3rd sideNeed to know: length of 3rd side

What is the Pythagorean Theorem?

a2 b2 c2

Where a and b are legs and c is the hypotenuse

Using aUsing a22 + b + b22 = c = c22

Looking for length ofLooking for length of

the hypotenusethe hypotenuse aa22 + b + b22 = c = c22

151522 + 20 + 2022 = c = c22

225 + 400 = c225 + 400 = c22

625 = c625 = c22

625 c2

25 c

Using aUsing a22 + b + b22 = c = c22

Looking for length ofLooking for length of

A legA leg

aa22 + b + b22 = c = c22

6622 + b + b22 = 10 = 1022

36 + b36 + b22 = 100 = 100 -36 -36-36 -36 bb22 = 64 = 64 b = 8b = 8

Applying the Applying the Pythagorean TheoremPythagorean Theorem

A 15 foot ladder leans up against a A 15 foot ladder leans up against a building. The foot of the ladder is 5 building. The foot of the ladder is 5 feet from the base of the building. feet from the base of the building.

How high up the wall, to the nearest How high up the wall, to the nearest foot does the ladder reach?foot does the ladder reach?

Draw a picture:Draw a picture:

Solving the problemSolving the problem

aa22 + b + b22 = c = c22 Write formulaWrite formula xx22 + 5 + 522 = 15 = 1522 Substitute inSubstitute in xx22 + 25 = 225 + 25 = 225 Solve for xSolve for x - 25 -25- 25 -25 xx22 = 200 = 200

x = 14.142135 x = 14.142135 Check: How am ICheck: How am I to leave my answer?to leave my answer?

x 200

The ladder reaches 14 feet up the wall.

Tim rode 8 miles due north, then 3 miles Tim rode 8 miles due north, then 3 miles due east. How far, to the nearest mile, is due east. How far, to the nearest mile, is

Tim from where he started? Tim from where he started?

Draw a picture:Draw a picture:

Solve the problemSolve the problem

aa22 + b + b22 = c = c22

8822 + 3 + 322 = c = c22

64 + 9 = c64 + 9 = c22

73 = c73 = c22

73 cRemember: How am I suppose to leave my answer?C = 8.5440037

Tim is 9 miles from where he started.

Further ApplicationsFurther ApplicationsThe diagonals of a rhombus are 6 The diagonals of a rhombus are 6 cm and 8 cm. What is the length cm and 8 cm. What is the length

of each side of the rhombus?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:

aa22 + b + b22 = c = c22

3322 + 4 + 422 = c = c22

9 + 16 = c9 + 16 = c22

25 = c25 = c22

25 c

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 As seen in the accompanying diagram, a person can travel from NYC to Buffalo by going north can travel from NYC to Buffalo by going north 170 miles to Albany and then west 280 miles to 170 miles to Albany and then west 280 miles to

BuffaloBuffalo. . A) A) If a highway is built to connect NYC and If a highway is built to connect NYC and

Buffalo, how many miles would be saved on the Buffalo, how many miles would be saved on the trip?trip?

B) With gas prices at $3.10 and a vehicle that B) With gas prices at $3.10 and a vehicle that gets 18 mpg, how much money would be saved gets 18 mpg, how much money would be saved

roundtrip, if the new highway was traveled roundtrip, if the new highway was traveled instead of the old route?instead of the old route?

Find length of new highwayFind length of new highway

Buf Albany

New York City

170 miles

280 miles

???

a2 + b2 = c2

2802+1702=c2

107300 m= c2

107300 c2

327.566= c

Did I answer question?

How many miles would be saved?

Old Distance: 280 + 170 = 450Old Distance: 280 + 170 = 450 New Distance: 327.566New Distance: 327.566 Saved Miles: 122.4 or 122 milesSaved Miles: 122.4 or 122 miles

How much money can be How much money can be saved?saved?

Saved Miles: 122 miles x 2 = 244Saved Miles: 122 miles x 2 = 244 Cost to drive one mile (gas):Cost to drive one mile (gas): $3.10 divided by 18. ($0.1722…)$3.10 divided by 18. ($0.1722…)

Cost to drive 244milesCost to drive 244miles $ 0.1722 times 244$ 0.1722 times 244 Saved: $42.02Saved: $42.02

Wrapping It UPWrapping It UP

The Pythagorean Theorem The Pythagorean Theorem can be used only on can be used only on _____triangles._____triangles.

When should the Pythagorean When should the Pythagorean Theorem be used?Theorem be used?

What should be done first What should be done first when solving a word when solving a word problem involving the problem involving the Pythagorean Theorem?Pythagorean Theorem?

What must be done before What must be done before writing the answer to a writing the answer to a Pythagorean Theorem Pythagorean Theorem problem?problem?

RightRight

When the length of 2 sides are When the length of 2 sides are known and the length of 3rd side known and the length of 3rd side is neededis needed

Draw and label triangleDraw and label triangle

Check to see whether the answer Check to see whether the answer should be rounded or notshould be rounded or not