The Pythagorean Theorem_1

8/10/2019 The Pythagorean Theorem_1

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The Pythagorean theorem

Standard: DemonstrationCreated by: Velumani. R

Description

Introduction to Demonstrate the Pythagorean theorem, and itss applications with examples oPythagorean theorem, examples and practice problems.

Introduction

!nimation: Show a right triangle and display it"s side as #$eg %&, #$eg '& and #(ypotenuse&.

Instructions to animation: )irst show this triangle as a larger one. (ighlight the parts #$eg %&, #$eg '&and #(ypotenuse&. !ter highlighting the parts, mo*e this triangle to the right side and ma+e it small.he igure should remain as it is showing the parts. I possible, include any sounds.

here is a special relationship among the lengths o the sides o any right triangle. his well-+nown

relationship is called the Pythagorean theorem, named or ater Pythagoras, a ree+ mathematician.his concept is oten used to calculate distances in real-lie problems situations.

he illustration gi*en below will *isuali/es the conceptclearly.:a+e a right- angled triangle o sides 0, 1 and 2 units.!nimation: 3o*e the right triangle gi*en abo*e and mar+ thesides as 0, 1 and 2 as shown here 4a+e o #$eg %&, #$eg '&and #(ypotenuse&. )irst display the right triangle alone.

$et"s Ddraw the s5uares on each side o the triangle.!nimation: Show the s5uares on the sides o the triangle. hisshould be shown at the center o the slide as a larger pictureand then the animation should mo*e to the right and should become smaller.!rea o the s5uare on the largest side hypotenuse 6 '2!rea o the s5uare on leg %6 7!rea o the s5uare on leg '6 %8

!nimation: (ighlight the respecti*e s5uares and their *alues while displaying '2, 7 and %8.



!nimation: Show the s5uares coming one-by-one rom the igure atright.

9otice that '2 6 7 %8.

;r 2

'

6 0

'

1

'

4<nd o slide %=

!nimation: !gain show a right triangle in a larger picture with the sides named a, b and c in the centero the slide. (ighlight the sides.

Suppose the sides are named a, b, and c , as shown in the igure, we get the relation c' 6 a

2 + b

2.hat is, the area o the largest s5uare e5uals the sum o the areas o the two smallerother two s5uares.his relationship is true or any right triangle.

The Pythagorean theorem

In a right triangle, the s5uare o the length o the hypotenuse is e5ual to the sum o the s5uares o thelengths o the legs.

4i*e bac+ground color to the statement=

Statement and pProof of the Pythagorean theorem

Statement

In a right triangle, the s5uare o the length o the hypotenuse is e5ual to the sum o the s5uares o thelengths o the legs.

Given

Right triangle BCA with leg lengths a andb and hypotenuse c.

Prove

c' 6 a' b'.

Plan

Draw a perpendicular CP to thehypotenuse.



Proof

9o Statements Reasons

% Draw altitude CP to the hypotenuse!nimation: Show CP

)rom a point not on a line, exactly one > can bedrawn to the line.

' c

a= a

x ; c

b=

b

yhe length o each leg is the geometric mean between the length o its ad?acent segment o thehypotenuse and the entire hypotenuse.

0 cx 6 a'@ cy 6 b' 3eans A extremes property

1 cx cy 6 a' b' !ddition property

2 c4 x y= 6 a' b' Distributi*e property

8 c' 6 a' b' Substitution property

Conclusion

I a right triangle has legs o lengths a and b and hypotenuse o length c, then c' 6 a' b'.

Real-life example

! ladder is leaning against a wall. Its base is positioned 8 eet rom the wall. )ind the length o theladder i the top o the ladder rests against the wall at the height o B eet rom the ground.Solutionhe igure at right illustrates the situation.!nimation: Show the picture shown at righthe wall is at a right angle to the ground, the ladder is the hypotenuse and the legs measure 8 eet and Beet. as shown in the right triangle at igure '.

!nimation: (ighlight the right angle in ig% bring it as a right triangle and show all the partsin the ig' below. It should come rom igure -%.y the Pythagorean theorem,c' 6 a' b'.c' 6 8' B'

c' 6 08 81c 6 %DDSo the length o the ladder is % eet.



(ere is an acti*ity.:Drag the s5uares rom $eg % and $eg to' to ill up the empty s5uares on the hypotenuse and see i thereare enough number o small s5uares.!nimation: he small s5uares on the leg % and leg ' must be illed s5uares. <ach should it into theempty s5uares on hypotenuse.

!llow the student to drag the small s5uares rom 0x0 and 1x1 s5uare to 2x2 s5uare. I the studentcompletes illing them display the below lines. $et the student drag our small s5uares at a time.

he s5uares on both the legs could ill all the blan+ s5uares on the hypotenuse completely. EhyF hePythagorean theorem wor+s here. So the triangle must be a right triangle.



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The Pythagorean Theorem_1