2. Angle bisection

VC2M9SP03: level 9: Design, test and refine algorithms involving a sequence of steps and decisions based on geometric constructions and theorems; discuss and evaluate refinements

  • creating and testing algorithms designed to construct or bisect angles, using pseudocode or flow charts

2.1. Angle bisector

Here is simple directions for an algorithm to construct an angle bisector:

  • Input: An angle with vertex O and rays OA and OB

  • Step 1: Set the compass width to any convenient length and place the compass point at O

  • Step 2: Draw an arc that intersects both rays OA and OB and label the intersection points as P and Q

  • Step 3: Without changing the compass width, place the compass point at P

  • Step 4: Draw an arc above the Angle

  • Step 5: Without changing the compass width, place the compass point at Q

  • Step 6: Draw an arc above the angle that intersects the arc drawn from P at R

  • Step 7: Use the straightedge to draw a line segment from O through R

  • Step 8: OR is the angle bisector

Here is a diagram which illustrates the angle bisection.
../_images/angle_bisector.png
Here is the python to draw the angle bisection.
  1import turtle
  2import math
  3import random
  4
  5def label_point(t, label, penc="black"):
  6    """Write a label next to a turtle's position.
  7
  8    Args:
  9        t (turtle.Turtle): The turtle object.
 10        label (str): The text to write.
 11        penc (str): The color of the text. Defaults to "black".
 12    """
 13    t.penup()
 14    t.setx(t.xcor() - 12)
 15    t.sety(t.ycor() + 2)
 16    t.pendown()
 17    t.pencolor(penc)
 18    t.write(label, font=("Arial", 12, "normal"))
 19
 20# Define a function to move the turtle to a point (x, y) with the penup
 21def move_to(t, point):
 22    """Move the turtle to a given point without drawing.
 23
 24    Args:
 25        t (turtle.Turtle): The turtle object.
 26        point (tuple): The coordinates of the point as (x, y).
 27    """
 28    t.penup()
 29    t.goto(point)
 30
 31def move_arc(t, centre=(0, 0), angle=0, radius=10, extent=360):
 32    """Move the turtle along an arc with a given centre, angle and radius.
 33
 34    Args:
 35        t (turtle.Turtle): The turtle object.
 36        centre (tuple): The coordinates of the centre of the arc as (x, y). Defaults to (0, 0).
 37        angle (int): The angle of the initial position of the turtle relative to the centre. Defaults to 0.
 38        radius (int): The radius of the arc. Defaults to 10.
 39        extent (int): The angle of the arc that the turtle moves along. Defaults to 360.
 40    """
 41    move_to(t, centre)
 42    t.seth(angle)
 43    t.fd(radius)
 44    t.seth(angle + 90)
 45    t.circle(radius, extent=extent)
 46
 47
 48def draw_line(t, point, length, direction, penc="black"):
 49    """Draw a straight line from a given point with a given length, direction and colour.
 50
 51    Args:
 52        t (turtle.Turtle): The turtle object.
 53        point (tuple): The coordinates of the starting point as (x, y).
 54        length (int): The length of the line.
 55        direction (int): The direction of the line in degrees.
 56        penc (str): The color of the line. Defaults to "black".
 57    """
 58    move_to(t, point)
 59    t.setheading(direction)
 60    t.pendown()
 61    t.pencolor(penc)
 62    t.forward(length)
 63
 64
 65def draw_centered_arc(t, centre=(0, 0), angle=0, radius=10, extent=360, penw=1, penc="black"):
 66    """Draw an arc with a given centre, angle, radius, extent, pen width and color.
 67
 68    Args:
 69        t (turtle.Turtle): The turtle object.
 70        centre (tuple): The coordinates of the centre of the arc as (x, y). Defaults to (0, 0).
 71        angle (int): The angle of the initial position of the turtle relative to the centre. Defaults to 0.
 72        radius (int): The radius of the arc. Defaults to 10.
 73        extent (int): The angle of the arc that the turtle draws. Defaults to 360.
 74        penw (int): The width of the pen. Defaults to 1.
 75        penc (str): The color of the pen. Defaults to "black".
 76    """
 77    move_to(t, centre)
 78    t.seth(angle)
 79    t.pensize(penw)
 80    t.pencolor(penc)
 81    t.fd(radius)
 82    t.pd()
 83    t.seth(angle + 90)
 84    t.circle(radius, extent=extent)
 85    for i in range(int((360-extent)/4)): # Draw a dot at each point on the circle
 86        t.pd()
 87        t.circle(radius, extent=1)
 88        t.pu()
 89        t.circle(radius, extent=3)        
 90    
 91
 92def angle_bisector(t, size, angle):
 93    # Input: An angle with vertex O and rays OA and OB
 94    # Draw the rays OA and OB first
 95    ray_length = 1.1*size  # This is the dsdie lengths
 96    angle_AOB = angle  # This is the angle between OA and OB=
 97    O = (0, 0) 
 98    label_point(t, "O", "red")
 99    draw_line(t, O, ray_length, 0, penc="black")  # Draw the ray OA
100    label_point(t, "A", "black")
101    draw_line(t, O, ray_length, angle_AOB, penc="black")  # Draw the ray OB
102    label_point(t, "B", "black")
103
104    # Step 1: Set the compass width to any convenient length and place the compass point at O
105    compass_width = int(size/2)
106
107    # Step 2: Draw an arc that intersects both rays OA and OB and label the intersection points as P and Q
108    move_to(t, (compass_width, 0))
109    P = t.pos()
110    t.dot(5, "blue")
111    label_point(t, "P", "blue")
112
113    draw_centered_arc(
114        t,
115        centre=O,
116        angle=0,
117        radius=compass_width,
118        extent=angle_AOB,
119        penw=1,
120        penc="red",
121    )
122
123    move_to(t, P)
124    move_arc(t, centre=(0, 0), angle=0, radius=compass_width, extent=angle_AOB)
125    Q = t.pos()
126    t.dot(5, "blue")
127    label_point(t, "Q", "blue")
128
129    # Step 3: Without changing the compass width, place the compass point at P
130    move_to(t, P)
131
132    # Step 4: Draw an arc above the angle
133    draw_centered_arc(
134        t,
135        centre=P,
136        angle=0,
137        radius=compass_width,
138        extent=(1.2*angle_AOB),
139        penw=1,
140        penc="blue",
141    )
142
143    # Step 5: Without changing the compass width, place the compass point at Q
144    move_to(t, Q)
145
146    # Step 6: Draw an arc above the angle that intersects the arc drawn from P at R
147    draw_centered_arc(
148        t,
149        centre=Q,
150        angle=angle_AOB,
151        radius=compass_width,
152        extent=-(1.2*angle_AOB),
153        penw=1,
154        penc="blue",
155    )
156
157    move_to(t, P)
158    move_arc(t, centre=P, angle=0, radius=compass_width, extent=angle_AOB)
159    R = t.pos()
160    t.dot(5, "green")
161    label_point(t, "R", "green")
162
163    # Step 7: Use the straightedge to draw a line segment from O through R
164    move_to(t, O)
165    draw_line(t, O, 1.5*t.distance(R[0], R[1]), t.towards(R), penc="green")
166    # Step 8: OR is the angle bisector
167
168
169
170def main():
171    SCREEN_WIDTH = 800
172    SCREEN_HEIGHT = 600
173    SIZE = 220
174    ANGLE = random.randint(40, 90)
175
176    # Set up the turtle screen
177    s = turtle.Screen()
178    s.bgcolor("white")
179    s.title("angle bisector")
180    s.setup(width=SCREEN_WIDTH, height=SCREEN_HEIGHT, startx=0, starty=0)
181    s.tracer(0, 0)
182
183    # Create a turtle object
184    t = turtle.Turtle()
185    t.speed(0) # Set the turtle's speed to the fastest
186    t.ht()
187
188    angle_bisector(t, size=SIZE, angle=ANGLE)
189
190    s.update()
191    s.exitonclick()
192
193
194if __name__ == "__main__":
195    main()