The Pythagorean Theorem

Slideshow 37, MathematicsMr. Richard Sasaki, Room 307

ObjectivesObjectives• Understand the relationship of the areas of

the squares about a right-angled triangle• Understand and derive the Pythagorean

Theorem• Be able to implement the theorem in simple

cases

The Right-Angled Triangle The Right-Angled Triangle Let’s review some vocabulary.

HypotenuseLegs

As you know, there is a relationship between the legs on a triangle and its hypotenuse. Before that, consider the diagram below.

A

B

CLet A, B and C be the areas of the given squares. What is the relationship between their areas? A + B = CLet’s try and prove this! Good luck!

Proof – Part 1Proof – Part 1A

B C

Consider some where . Squares ACIH, BCED and ABFG exist as shown.

G

F

HI

D E

Consider where , L exists on and K exists on .

K

LDraw and so and exist.

As , C, A and G are collinear. B, A and H are also collinear. .As Also, ) and by SAS.

As both have base and height , is half the area of rectangle BDLK.

Proof – Part 2Proof – Part 2A

B C

G

F

HI

D E

K

L

Also for the same reasons, as C, A and G are collinear, square BAGF is twice the area of . As , square BAGF has the same area as rectangle BKLD.For the same reasons, square AHIC has the same area as KCEL.Adding both results, we get the total area of squares BAGF and AHIC being the total area of rectangles BKLD and KCEL, which is equal to the area of square BCED. The total area of squares ABFG and ACIH is equal to the area of square BCED.

AnswersAnswers1. 2. 3. 4. 5. 6. 7.

The Pythagorean TheoremThe Pythagorean TheoremThe Pythagorean Theorem, has hundreds of proofs.My favourite proof is Garfield’s proof. Consider a trapezium ABCD with two right angles as shown.

A

B C

D

Split the trapezium into 3 triangles, two of which are congruent with legs and hypotenuse . The area for the trapezium exists where . As both triangles are congruent, we can label the angles as shown about . Hence .𝑎

𝑎

𝑏

𝑏 𝑐

𝑐𝛼

𝛼

90−𝛼

90−𝛼

E

Area of TrianglesCombining the areas,

⇒𝑐2=𝑎2+𝑏2⇒𝑎2+𝑏2=𝑐2

The Pythagorean TheoremThe Pythagorean TheoremYou will have an opportunity to learn more proofs in the Winter Homework. You should know at least one.

𝑎

𝑏

𝑐Pythagorean Theorem: 𝑎2+𝑏2=𝑐2

Note: and represent the legs and represents the hypotenuse. Also, the triangle must have a right-angle.

We can use the theorem to find missing lengths.ExampleIf and , calculate .52+122=𝑐2⇒169=𝑐2⇒𝑐=¿13

Note: No length can be negative!

AnswersAnswers

1520

37

√17622632

5√3

Surds and PythagorasSurds and PythagorasAs you saw in the last question, remember to simplify surds when you can!Example

4

3√2

𝑎Calculate .

𝑎2+𝑏2=𝑐2

Don’t forget, this is (the hypotenuse).

⇒ 42+(3√2 )2=𝑎2

If it confuses you, don’t write it (unless you are told to)!

⇒16+18=𝑎2⇒𝑎2=34⇒𝑎=√34

This doesn’t simplify.

AnswersAnswers

48 4√5 20±18 48 6 √5

2√112√103