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Geometric Construction 2 Circles Arcs Polygon
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
DETERMINING THE CENTER OF AN ARC
2. Draw another arbitrary line, connected to point B with the other endpoint on the circumference labeled as C.
A
B
C
DETERMINING THE CENTER OF AN ARC
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.
DETERMINING THE CENTER OF AN ARC
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
DRAWING A CIRCLE/ARC THROUGH THREE POINTS
1. With straight lines, lightly connect points A to B, and B to C.
A
B
C
DRAWING A CIRCLE/ARC THROUGH THREE POINTS
2. Using the method outlined for bisecting a line, bisect lines A-B and B-C.
A
B
C
DRAWING A CIRCLE/ARC THROUGH THREE POINTS
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
DRAWING A CIRCLE/ARC THROUGH THREE POINTS
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
RECTIFYING AN ARC LENGTHS
1. Find the center of the arc (see procedure for finding the center of a circle).
A
B
C
X
RECTIFYING AN ARC LENGTHS
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
RECTIFYING AN ARC LENGTHS
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
RECTIFYING AN ARC LENGTHS
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
RECTIFYING AN ARC LENGTHS
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
SETTING OFF A GIVEN LENGTH ALONG AN ARC
1. Find the center of the given arc (see steps in finding the center of an arc).
X
J
F
SETTING OFF A GIVEN LENGTH ALONG AN ARC
2. Connect the center to either of the endpoints. Draw line perpendicular to line XF and tangent to the given arc.
X
J
F
SETTING OFF A GIVEN LENGTH ALONG AN ARC
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’
SETTING OFF A GIVEN LENGTH ALONG AN ARC
4. Divide line A’F into four equal segments. Label the points as 1, 2, and 3.
X
J
F
A’
1
2
3
SETTING OFF A GIVEN LENGTH ALONG AN ARC
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
SETTING OFF A GIVEN LENGTH ALONG AN ARC
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
DRAWING AN EQUILATERAL TRIANGLE
1. Copy the given length.
A B
DRAWING AN EQUILATERAL TRIANGLE
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
DRAWING AN EQUILATERAL TRIANGLE
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
Location of the triangle
Hypotenuse
DRAWING A TRIANGLE GIVEN THE HYPOTENUSE AND A GIVEN LEG
1. Using the length of the given hypotenuse as diameter, draw a semi-circle.
A B
DRAWING A TRIANGLE GIVEN THE HYPOTENUSE AND A GIVEN LEG
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
DRAWING A TRIANGLE GIVEN THE HYPOTENUSE AND A GIVEN LEG
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
Location of the triangle
DRAWING A TRIANGLE GIVEN THE LENGTH OF THE THREE SIDES
1. Layout Side A in the desired position.
1 2
DRAWING A TRIANGLE GIVEN THE LENGTH OF THE THREE SIDES
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
DRAWING A TRIANGLE GIVEN THE LENGTH OF THE THREE SIDES
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
DRAWING A TRIANGLE GIVEN THE LENGTH OF THE THREE SIDES
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
Center at point 2 with arbitrary radius R1
INSCRIBING A CIRCLE INSIDE TRIANGLE ABC
1. Bisect angle B by line BE intersecting line AD at point O.
B
C
A
D O 4
3
Center at point 3 with arbitrary radius R2
Center at point 4 with arbitrary radius R2
E
INSCRIBING A CIRCLE INSIDE TRIANGLE ABC
2. Draw line FG through point O perpendicular to side AB at point H.
B
C
A
D O
H
E
INSCRIBING A CIRCLE INSIDE TRIANGLE ABC
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
CIRCUMSCRIBING A CIRCLE AROUND TRIANGLE ABC
1. Draw a perpendicular bisector (Line DE) to side AB.
CIRCUMSCRIBING A CIRCLE AROUND TRIANGLE ABC
2. Draw a perpendicular bisector (Line FH) to side BC intersecting Line DE(first bisector) at point O.
O
CIRCUMSCRIBING A CIRCLE AROUND TRIANGLE ABC
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.
INSCRIBING AN EQUILATERAL TRIANGLE IN A CIRCLE OF RADIUS R
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.
INSCRIBING AN EQUILATERAL TRIANGLE IN A CIRCLE OF RADIUS R
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.
INSCRIBING AN EQUILATERAL TRIANGLE IN A CIRCLE OF RADIUS R
O
D
A C
B
4. Connect point D to points B and C to complete the triangle.
INSCRIBING AN EQUILATERAL TRIANGLE IN A CIRCLE OF RADIUS R
O
D
A C
B
DRAWING A SQUARE WITH SIDE AB GIVEN
• Given: Length of side AB
B A
Location of the square
DRAWING A SQUARE WITH SIDE AB GIVEN
1. Draw side AB in the desired position. Construct line BE perpendicular to side AB and originating from point B.
A B
E
DRAWING A SQUARE WITH SIDE AB GIVEN
2. Using point B as center and AB as radius, draw an arc cutting line BE at point C.
A B
E
C
DRAWING A SQUARE WITH SIDE AB GIVEN
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
DRAWING A SQUARE WITH SIDE AB GIVEN
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
DRAWING A SQUARE INSIDE A CIRCLE
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
DRAWING A SQUARE INSIDE A CIRCLE
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
DRAWING A SQUARE INSIDE A CIRCLE
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
INSCRIBING A PENTAGON INSIDE A CIRCLE
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
INSCRIBING A PENTAGON INSIDE A CIRCLE
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
INSCRIBING A PENTAGON INSIDE A CIRCLE
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
INSCRIBING A PENTAGON INSIDE A CIRCLE
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.
DRAWING REGULAR POLYGON(Method 1)
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.
DRAWING REGULAR POLYGON(Method 1)
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.
DRAWING REGULAR POLYGON(Method 1)
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
DRAWING REGULAR POLYGON (Method 2)
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
DRAWING REGULAR POLYGON (Method 2)
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
DRAWING REGULAR POLYGON (Method 2)
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
