Geometric Construction 2

Geometric Construction 2

Circles

Arcs

Polygon

DETERMINING THE CENTER

•Arc

•Circle

DETERMINING THE CENTER OF AN ARC/CIRCLE

• Given: an arc

DETERMINING THE CENTER OF AN ARC

1. Draw an arbitrary line with endpoints on the circumference of the circle. Label the endpoints of the chord as A and B.

A

B


2. Draw another arbitrary line, connected to point B with the other endpoint on the circumference labeled as C.

A

B

C


3. Using the method outlined for bisecting a line, bisect lines A-B and B-C.

B

C

A Center at A Radius greater than one-half AB

Center at B Radius greater than one-half AB.

Center at B Radius greater than one-half BC.

Center at C Radius greater than one-half BC.


4. Locate point X where the two extended bisectors meet. Point X is the exact center of the circle.

B

C

A

X

Drawing a circle/arc

through three points

DRAWING A CIRCLE/ARC THROUGH THREE POINTS

• Given: Three points in space at random: A, B, and C.

A

B

C


1. With straight lines, lightly connect points A to B, and B to C.

A

B

C


2. Using the method outlined for bisecting a line, bisect lines A-B and B-C.

A

B

C


3. Locate point X where the two extended bisectors meet. Point X is the exact center of the arc or circle.

A

B

C X


4. Using X as center and radius equal to XA (or XB or XC), draw a/an circle/arc. The circle/arc drawn passed through the three given points.

A

B

C X

RECTIFYING AN

ARC LENGTHS

RECTIFYING AN ARC LENGTHS

•Given: an arc


1. Find the center of the arc (see procedure for finding the center of a circle).

A

B

C

X


2. Form the longest chord and divide it into two (see procedure on how to bisect a line). Connect either of the arc’s endpoints to its center.

A

B

X

O

C

2

1


3. Extend the chord. The length of the extension must be equal to O2 or one-half of the chord 12.

A

B

X

O

C

3

2

1

Line O2 = Line 23


4. Draw a line perpendicular to the line connected to the arc’s center and tangent to the circle.

A

B

X

O

C

3

1

2


5. Using point 3 as center and radius equal to line 13, strike an arc intersecting the tangent line at point 4.

A

B

X

O

C

3

2

1

4

Line C4 is the rectified

length of arc 12.

setting off a given length

along an arc

SETTING OFF A GIVEN LENGTH ALONG AN ARC

• Given: Line AB and an arc JF

A B

J

F


1. Find the center of the given arc (see steps in finding the center of an arc).

X

J

F


2. Connect the center to either of the endpoints. Draw line perpendicular to line XF and tangent to the given arc.

X

J

F


3. Layout the length of line AB in the tangent line (recall steps in transferring a line). Label the intersection as A’.

X

Length of line AB

J

F

A’


4. Divide line A’F into four equal segments. Label the points as 1, 2, and 3.

X

J

F

A’

1

2

3


4. Using point 1 as center and radius equal to line 1A’, strike an arc intersecting the given arc. Label the intersection as C.

X

J

F

A’

1

2

3 C


4. Arc FC is approximately equal to line AB.

X

J

F

A’

1

2

3 C

triangles

DRAWING AN EQUILATERAL TRIANGLE

• Given: length of the sides

A B

Location of the triangle


1. Copy the given length.

A B


2. Using point A as center and radius equal to the length of the given side, draw an arc. Repeat the step, using B as center.

A B

Center at A Radius equal to AB Center at B

Radius equal to AB


3. Locate Point 1 where the arcs intersect. Connect the endpoints to Point 1.

A B

1

DRAWING A TRIANGLE GIVEN THE HYPOTENUSE AND A GIVEN LEG

• Given: - length of one side - length of hypothenuse

A

B

B

C


Hypotenuse


1. Using the length of the given hypotenuse as diameter, draw a semi-circle.

A B


2. Using one endpoint of the hypotenuse as center and the length of the side BC as radius, draw an arc intersecting the semi-circle at point C.

C

A B


3. Connecting point C with endpoints A and B establishes the desired Right Triangle ABC

C

A B

DRAWING A TRIANGLE GIVEN THE LENGTH OF THE THREE SIDES

• Given: length of three sides

1

1

2

3

3 2

Side A

Side C

Side B



1. Layout Side A in the desired position.

1 2


2. Using endpoint 1 of side A as center and the length of side B as radius, draw an arc above side A.

1 2

Center at 1 Radius equal to side B


3. Using endpoint 2 of side A and the length of side C as radius, draw a second arc intersecting the first arc at point 3.

1 2

Center at 2 Radius equal to side C

3


4. Connecting point 3 with points 1 and 2 establishes Triangle ABC.

1 2

3

INSCRIBING A CIRCLE INSIDE TRIANGLE

• Given: Triangle ABC

B

C

A

INSCRIBING A CIRCLE INSIDE TRIANGLE ABC

1. Bisect angle A by line AD extending this beyond the middle of the triangle.

B

C

A

D

1

2

Center at point 1 with arbitrary radius R1



1. Bisect angle B by line BE intersecting line AD at point O.

B

C

A

D O 4

3



E


2. Draw line FG through point O perpendicular to side AB at point H.

B

C

A

D O

H

E


3. Using point O as center and radius equal to OH, draw the desired circle.

B

C

A

D O E

H

CIRCUMSCRIBING A CIRCLE AROUND TRIANGLE ABC

• Given: Triangle ABC


1. Draw a perpendicular bisector (Line DE) to side AB.


2. Draw a perpendicular bisector (Line FH) to side BC intersecting Line DE(first bisector) at point O.

O


3. Using point O as center and OC (or OB) as radius , draw the desired circumscribed circle.

O

INSCRIBING AN EQUILATERAL TRIANGLE IN A CIRCLE OF RADIUS R

• Given: Radius of circle R.

R

1. Using the given radius, draw circle O.


O R

2. Designate any point A in the circumference of the circle, point D is located at the opposite end of the diameter line.


O

D

A

3. Using point A as center and radius R equal to the radius of the circle, draw an arc cutting the circumference of the circle at point B and at point C.


O

D

A C

B

4. Connect point D to points B and C to complete the triangle.


O

D

A C

B

DRAWING A SQUARE WITH SIDE AB GIVEN

• Given: Length of side AB

B A

Location of the square


1. Draw side AB in the desired position. Construct line BE perpendicular to side AB and originating from point B.

A B

E


2. Using point B as center and AB as radius, draw an arc cutting line BE at point C.

A B

E

C


3. Using points A and C as centers and the same radius in both operations, draw two arcs intersecting each other at point D.

A B

E

C D


4. Connect point C to point D and point A to point D.

A B

E

C D

DRAWING A SQUARE INSIDE A CIRCLE

• Given: Radius of circle

R


1. Draw the circle with point E as center. Draw line AB through point E cutting the circle at point G and H.

E

B

A

G

H

R


2. Draw line CD perpendicular to line AB passing through point E and cutting the circle at points M and N.

E

B

A

G

H

C

D

N

M


3. Connect points G to M, M to H, H to N, and N to G.

E

B

A

G

H N

M

C

D

DRAWING A RECTANGLE

• Given: length of diagonal and length of one side

B D

Diagonal

B C

DRAWING A RECTANGLE

1. Draw the diagonal BD and bisect it at point O. Using point O as center, draw a circle passing through point B and point D. Line BD is a diameter.

B D O

DRAWING A RECTANGLE

2. Using points B and D as centers, and length of side BC as radius, draw two arcs cutting the circle at point C and point A.

B D

A

C

O

DRAWING A RECTANGLE

3. Connect point B to point C, C to D, D to A, and A to B to complete the rectangle.

B D O

A

C

INSCRIBING A PENTAGON INSIDE A CIRCLE

• Given: radius of the circle

R


1. Draw two diameters of the circle which are perpendicular to each other, cutting the circumference of the circle at points A, L, M, N.

O

R

A

N

M

L


2. Bisect radius OL at point P, from point P and using the distance between point P and point A as radius, draw an arc cutting radius ON at point X.

A

N

M

L

R

O

P X


3. From point A and using the distance between point A and point X as radius, draw a second arc cutting the circle at point B.

A

N

M

L

R

O

P X

B


4. Draw line AB and use its length to determine points C, D and E around the circumference of the circle. Connect the points.

A

N

M

L

R

O

P X

B E

D C

INSCRIBING A REGULAR POLYGON INSIDE A GIVEN CIRCLE

• Given: radius of the circle n (number of sides) ex. n=6

R

1. Draw a circle and divide its diameter, line A-B, into n-parts (number of sides of the polygon). Label them 1-(n-1).

DRAWING REGULAR POLYGON(Method 1)

2 1 A 3 B 4 5

2. Using A (then B) as center and radius equal to line AB, draw an arc. Where the arcs intersect, locate point C.


2 1 A 3 B 4 5

C

3. Draw a line connecting point C to point 2 and extend the line. Locate point D where the extended line intersects the circle.


2 1 A 3 B 4 5

C

D

4. Connect points A and D. Using the length of line AD draw the other side of the polygon.


2 1 A 3 B 4 5

C

D

INSCRIBING A REGULAR POLYGON (Method 2)

• Given: length of one side

B C

DRAWING REGULAR POLYGON (Method 2)

1. Recall method in constructing a square given side AB.

A B


2. Draw the diagonals of the square. Label the intersection of the diagonal as 4. Point 4 is the center of the circle that can inscribe a square.

A B

4


3. Recall steps in constructing equilateral triangle. Label the intersection as 6. Point 6 is the center of the circle that can inscribe a hexagon.

A B

4

6


4. Connect point 4 and point 6. Bisect line 46. Label the midpoint as 5. Point 5 is the center of the circle that can inscribe a pentagon with sides equal to AB.

A B

4

6

5

Geometric Construction 2

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