My Ppt On Constructions

Constructions of various shapes using only compass and straightedge/ruler

Compiled by Pedup Dukpa (For third year Paro College of Education students)

Important questions to ask yourself: What is construction? Who invented this tool commonly used in geometry? Geometry? What? And why? When ….? Where…? How….?

TOOLS NEEDED COMPASS STRAIGHT EDGE PENCIL PAPER YOUR BRAIN (THE MOST IMPORTANT TOOL)

What do we mean by construction? the drawing of geometric items such as lines and circles using only a compass and straightedge. Very importantly, you are not allowed to measure angles with a protractor, or measure lengths with a ruler.

Compass

The compass is a drawing instrument used for drawing circles and arcs. It has two legs, one with a point and the other with a pencil or lead. You can adjust the distance between the point and the pencil and that setting will remain until you change it. Note: This kind of compass has nothing to do with the kind used find the North direction when you are lost. Straightedge A straightedge is simply a guide for the pencil when drawing straight lines. In most cases you will use a ruler for this, since it is the most likely to be available, but you must not use the markings on the ruler during constructions. If possible, turn the ruler over so you cannot see them.

Father of Geometry Euclid, the ancient Greek mathematician is the acknowledged to be the inventor of geometry. He did this over 2000 years ago, and his book "Elements" is still regarded as the ultimate geometry

Why did Euclid do it this way?  Why didn't Euclid just measure things with a ruler and calculate lengths so as to bisect them?

The Greeks could not do arithmetic. They had only whole numbers, no zero, and no negative numbers like the Roman numerals. In short, they could perform very little useful arithmetic. So, faced with the problem of finding the midpoint of a line, they could not do the obvious - measure it and divide by two. They had to have other ways, and this lead to the constructions using compass and straightedge. It is also the reason why the straightedge has no markings. It is definitely not a graduated ruler, but simply a pencil guide for making straight lines. Euclid and the Greeks solved problems graphically, by drawing shapes, as a substitute for using arithmetic.

Enough Intro.  Let’s get it started… with the topic...

Bisecting a Line Segment  Given the line segment extending from point A to point B, carry out the following steps to construct the perpendicular bisector.

 Step 1: Let r be greater than one-half of AB. Construct a circle of radius r at both points A and B.

 Step 2: Connect with a line segment the two points where the circles intersect.

 You're done! Line EF is the perpendicular bisector of Segment AB.

Investigation

Investigation

Angle Bisector Construction Aim: Divide an angle in half

Start

Angle Bisector A Aim: divide an angle in half

Compass point

Step1

Arc over both lines

Home

Angle Bisector Aim: divide an angle in half

Compass point

Step1

Arc over both lines

Home

Angle Bisector Aim: divide an angle in half

Compass point

Step1

Arc over both lines

Home

Angle Bisector B Aim: divide an angle in half Keep the compass the same

Arc into space

Compass point

Step1

Step2

Home

Angle Bisector Aim: divide an angle in half Keep the compass the same

Arc into space

Compass point

Step1

Step2

Home Home

Angle Bisector Aim: divide an angle in half Keep the compass the same

Arc into space

Compass point

Step1

Step2

Home

Angle Bisector C Aim: divide an angle in half Keep the compass the same Arc to cross Compass point

Step1

Step2

Step3

Home

Angle Bisector Aim: divide an angle in half Keep the compass the same Arc to cross Compass point

Step1

Step2

Step3

Home

Angle Bisector Aim: divide an angle in half

With this point Use a ruler to join this point…

0

Step1

1

Step2

2

3

4

Step3

5

6

7

Step4

8

11 12 0 1 9

Home

Angle Bisector Aim: divide an angle in half Angle bisector

The angle is now cut in half

1 2 1 2

Step1

Step2

Step3

Step4

The End

In circle

Home

In circle Do the angle bisectors of the three corners of a triangle meet at a point?

Home

In circle Do the angle bisectors of the three corners of a triangle meet at a point?

Like this?

Home

In circle Do the angle bisectors of the three corners of a triangle meet at a point?

OR this?

Home

In circle Do the angle bisectors of the three corners of a triangle meet at a point?

Home

Construction of in-circle of a triangle a

i

x x b

Draw the triangle abc ∠abc using circle arcs. Construct the bisector of ∠acb Construct the bisector of The bisectors meet at i the incentre of the triangle Using i as centre construct the incircle of the triangle abc

o o c

Construction of Circumcircle Steps of Construction of a triangle C

Construct a Δ ABC Bisect the side AB Bisect the side BC

o

The two lines meet at O From O Join B

A

B

Taking OB as radius draw a circumcircle.

Construction of parallel lines:

How To Construct Parallel Lines Given line k and an exterior point P Step 1:Draw an arbitrary line through point P, intersecting line k. Call the intersection point Q.

P

Q

k

Step 2:Center the compass at point Q and draw an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc

P

Q

k

Step 3:Set the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects line PQ. Mark the arc intersection point R.

P R

Q

k

Finally, join point P with the point R. Line PR is parallel to line k

P R R

Q

k

Construction of perpendicular lines: Homework (You may work in groups but please don’t directly copy your fren’s work… )

Types of triangle:  According to sides: Scalene Triangle  Isosceles Triangle  Equilateral Triangle  According to angles:  Acute Angled Triangle  Obtuse Angled Triangle\  Right Angled Triangle 

Note: There are actually 7 types.

I have attached the other notes as a document and not power point presentation (PPT) slides, mainly because I stop using PPT as suggested by one of your fren’s.