a Draw the first diameter and recall how to construct the perpendicular bisector of a segment.
B
b Recall how to construct an angle bisector.
C
c How many sides does the polygon have?
D
d Draw 6 equilateral triangles with the center of the circle as a common vertex.
A
a See solution.
B
b See solution.
C
c See solution.
D
d See solution.
Practice makes perfect
a With a compass, let's draw a circle.
Next, we draw a diameter of the circle. To do that, we must draw a segment that passes through the center and that has endpoints on the circle.
We now want to draw another diameter of the circle such that it is perpendicular to the one we drew above. To do that, we place the tip of the compass on A, open it greater than the radius of the circle, and draw an arc.
Next, with the same compass setting, we repeat the process but this time we place the compass tip on B.
Finally, we draw the segment that passes through the center and the intersection point of the two arcs drawn. It will be a diameter that is perpendicular to AB.
b In this part, we have to bisect each of the four right angles formed by the two diameters drawn in Part A. To do that, we place the compass tip on the center of the circle and draw an arc that intersects both diameters.
Let X and Y be the intersection points. Now, we place the compass tip on X and draw an arc above AB.
With the same compass setting, we place the compass tip on Y and draw a second arc that intersects the one we just drew.
The segment passing through the center of the circle and the intersection point between the arcs is a diameter that bisects two of the four right angles.
Doing a similar process, we can draw the fourth diameter which bisects the remaining two right angles.
c Let's connect the consecutive points where the diameters intersect the circle.
As we can see, we obtained a regular polygon with 8 sides. This is an octagon.
d Let's consider a circle and a point A on it.
Since a hexagon is formed by 6 equilateral triangles, we will draw the triangles with P as a common vertex. To do that, we place the compass tip on A and, with the same compass setting used before, we draw an arc that intersects the circle.
Let B the intersection point. We have that â–ł ABP is equilateral. Next, with the same compass setting, we place the compass tip on B and draw another arc that intersects the circle (not at A).
By repeating this process 3 more times, we will get a total of 6 points. By connecting them, we will obtain a regular hexagon (and 6 equilateral triangles).
