#### Transcript Slide 1

Theorem 14: [Theorem of Pythagoras] In a right-angled triangle the square of the hypotenuse is the sum of the squares of the other two sides. USE THE FORWARD AND THE BACK ARROWS ON THE KEYBOARD TO VIEW AND REWIND PROOF.

Given: Triangle ABC with |

BAC| = 90 O

B

To Prove: |BC| 2 = |AB| 2 + |AC| 2 Construction: Proof:

B

Draw a perpendicular from A to meet BC at D In the triangles ABC and ADC

D

90 o

D

90 o 90 o 90 o

A

|

A  A

90 o

C

BAC| = |

BCA| = |

ACD| Same angle

C 

∆ ABC and ∆ ADC are equiangular.



∆ ABC and ∆ ADC are similar. Theorem 13

C  | | BC AC | |  | AC | DC | | 

|AC| 2 = |BC| x |DC| ………….

Equation 1 © Project Maths Development Team

In the triangles ABC and ABD

B B  

90 o

D A

|

90 o

C A

BAC| = |

|

ABC| = |

ABD| Same angle ∆ ABC and ∆ ABD are equiangular.



∆ ABC and ∆ ABD are similar.

 | | BC AB | |  | | AB | BD |

Theorem 13



|AB| 2 = |BC| x |BD| ………….

Equation 2 2 Adding Equation 1 and Equation |AC| 2 = |BC| x |DC| … Equation 1 |AB| 2 = |BC| x |BD| … Equation 2 |AC| 2 + |AB| 2 = |BC| x |DC| + |BC| x |BD|

B A

90 o 90 o 90 o

D C

© Project Maths Development Team 2009



|AC| 2 + |AB| 2 = |BC| {|DC| + |BD|} |BC| is common

B

But |AC| 2 + |AB| 2 = |BC| {|DC| + |BD|} |BC| is common But DC| + |BD| = |BC|

 

|AC| 2 + |AB| 2 = |BC| x |BC| |AC| 2 + |AB| 2 = |BC| 2

|BC| 2 = |AB| 2 + |AC| 2 Q.E.D.

A

90 o 90 o 90 o

D C

© Project Maths Development Team 2009