angle bisectors on a grid

a ppt is here

this is part of a series of tasks involving a square dot grid and, in this case, expressing an angle as arc tan (fraction)
exploring when angle bisectors passing through integer (dot) grid points

these were used in several tasks, part of a session at Anne Watson and John Mason's house looking at the topic of 'angle'

you could use a few of the initial tasks to practice bisecting an angle with ruler, pencil and compasses

but the work continues, to see how pythagorean triples are involved and how to calculate the radius of the incircle for such triangles

angle bisectors

an intention of this task is that students use their angle bisecting skills to find and then possibly prove a relationship between two angles

a ppt is here

find the incentre and darw the circle just touching the three sides:

constructions

initial compass practice can be helpful for some students

the website sciencevsmagic is a most helpful resource: 40 construction tasks to attempt

Patrick Honner's blog contains the suggestion that students can create the angle bisectors (and so the incentre) by paper folding:

and the line bisectors
to construct the circumcentre














to practice the (bisector) constructions using compasses and a straight edge, students can be asked to construct all three angle bisectors of a triangle

with sufficient accuracy the lines will meet at a point, the 'incentre', and this circle can be drawn:

also for the perpendicular bisectors of the sides of a triangle - creating the 'circumcentre'

this probably needs a sheet of plain paper
possibly informing students that they can vary their compass radius for each side bisection

sometimes this circumcentre lies:

• inside the triangle
• outside it
• (borderline case) on one of the sides of a triangle

students can explore these three cases
noting the special case where the lines meet on one of the sides of the triangle


students can also be asked to bisect two joined ('hinged') lines - to find the centre of a circle that passes through the three points A, B and C (i.e. the circumcircle):

explore how three points determine a circle

usually...

further tasks, practicing these skills, could be to:

• construct a regular hexagon
• construct a 30 degree angle (and a 45 degree angle)
• construct an equilateral triangle

how to construct a parallel to a line (CD) through a given point (A) :

incircle of a right angled triangle

by considering areas, you can establish that the radius of the incircle is
ab/(a + b + c)

by considering equal (bits of) tangents you can also establish that the radius,
 r = 1/2 (b + a - c):

b + a - c  =  (e + g) + (e + f ) - (g + f)  =  2e  =  2r

for integer values of the incircle radius you need a pythagorean triple

with the (subset of) pythagorean triples generated from the shortest side being an odd number

3, 4, 5  has an incircle radius, r = 1
5, 12, 13  has r = 2 (property for shapes where the area value = perimeter value, 'equable')
7, 24, 25  has r = 3
9, 40, 41  has r = 4
etc.

using the (pythagorean triple) property that a chosen odd number ('a') when squared is broken up into two consective integers for 'b' and 'c', you can (using some reasonably complicated algebra for the first relationship above) show that this pattern continues, for ever and ever....

for such pythagorean triples (and the statements are also true in general), you can also establish that
3 must be a factor of either a or b (but not both)
4 is a factor of a or b
5 is a factor of a, b or c

alternative angle bisecting technique

several people (mainly Derek Ball) devised an interesting way of constructing an angle bisector in a meeting led by Steph Prestage and Pat Perks (then of Birmingham University)

draw two arcs from the meet point (the intersection of the two lines)

draw the cross diagonals of the resulting isosceles trapezium (or trapezoid)

join this crossover point to the start point

this same device (drawing two arcs) can be adapted to 'drop a perpendicular' to a line from a given point






Pat and Steph stress a view that it is helpful to use full circles rather than just arcs when constructing loci - to aid understanding

however, this can use (and needs) a lot of space...

thanks to Ed Southall