Welcome to lesson on constructing the perpendicular bisector of the sides
of a triangle.
The goals are to perform the construction and also define the
properties of the perpendicular bisectors
of the sides of a triangle. So in this diagram
the red lines are the perpendicular bisectors of the sides of the triangle.
So what that means is that these red lines and the sides of the triangle
from a right angle here, here,
and here. And that these cut each side
of the triangle into two congruent segments.
So this segment is congruent to this segment, this segment is congruent to this segment,
and this segment is congruent to this segment.
And notice how all three perpendicular bisectors
intersect in the same point. And this point is called the circumcenter.
So the circumcenter is the point of concurrency of the perpendicular bisectors
of the sides of the triangle. The circumcenter is also the center of a circle
that passes through the vertices of the triangle and that would be this green
circle
with its center here. So it follows that the circumcenter is
equidistants to the vertices. So if we drew a segment from the circumcenter
to each vertex, notice
each segment would be the radius of the green circle. And we say that the
green circle
circumscribes the triangle. Now let's go ahead and take a look at our construction.
We're going to need a compass and straightedge. And for this entire construction,
we're going to leave the radius of the compass the same. The only restriction is that we have to
make sure the radius of the compass
is more than half the length of the longest side of the triangle.
So this would be the longest side of the triangle. So let's go ahead and put the
point of the compass here, and then we'll put the pencil past the midpoint
maybe somewhere here. And so we'll leave this as the radius for the entire construction.
Now if we want to bisect this side of the triangle,
we'll swing an arc above and below this side with the point of the compass here.
So our arcs might look something like this.
Now leaving the radius of the compass the
same, we'll now put the point of the compass here,
and do the same. Swing an arc above and below this side.
It might look something like this.
And now the perpendicular bisector will pass through
the intersection points of these arcs. It would look like this.
So notice how it forms a right angle with this side
and it also cuts this side in a two congruent segments.
Now again leaving the radius the same, let's bisect
this shorter side. So we'll put the point of the compass here,
and we'll swing an arc to the left and right of this side. So the two arcs might look
something like this.
Then we'll put the point of compass
at this vertex and do the same. Swing an arc to the right
and swing an arc to the left. And the intersection points of these arcs
will be on our perpendicular bisector.
Again we have a right angle here,
and two congruent segments.
And now we have one more side to bisect, so we'll put the point of the compass here.
And swing an arc to left and right of this side, again with the same radius.
So it might look something like this, and something like this.
Move the point of the compass to this vertex and do the same.
Swing an arc to the left and swing an arc to the right.
And so this last perpendicular bisector will pass through this point of intersection,
and this point of intersection.
So a right angle here and two congruent segments.
My construction is little bit off, but you can see that the
perpendicular bisector should intersect here at the circumcenter.
And we could construct a circle with the center here that would pass through the three
vertices of the triangle.
I hope you found this useful.