Video Transcript
Draw the isosceles triangle 𝐴𝐵𝐶,
where 𝐴𝐵 equals 𝐴𝐶. Using a compass, bisect line
segment 𝐵𝐶 at point 𝐷. Find the measure of angle
𝐴𝐷𝐶.
We are told to begin this problem
by drawing an isosceles triangle, which we recall is a triangle with two equal
sides. Because we have 𝐴𝐵 equals 𝐴𝐶,
then we know that the sides 𝐴𝐵 and 𝐴𝐶 will be the two equal-length sides. And so, we can draw a triangle
something like this, labeled so that we do have sides 𝐴𝐵 and 𝐴𝐶 congruent.
Next, we are told to bisect line
segment 𝐵𝐶, which will be the horizontal line in the figure below. To bisect means to cut into two
equal pieces. But we aren’t going to measure the
line segment and then find two equal pieces because we are told to use a compass,
which is one of these tools. This may often be called a pair of
compasses, depending on where you live.
A compass has a sharp pointed end,
which is the end we place on a point on the page, and the pencil allows us to create
circles or arcs. A good tip is that it’s always best
to use a sharp pencil when using a compass and make sure that the screw of the
compass is tight enough so that the legs of the compass don’t move too easily.
We can consider the line segment
𝐵𝐶 that we need to bisect. The first step that we take is that
we set the radius of the compass to be greater than half the length of line segment
𝐵𝐶. So, we open up the compass so that
it’s further than halfway along the line segment 𝐵𝐶 from one of the endpoints. The second step is to trace the
arcs of two circles centered at 𝐵 and 𝐶. To do this, we begin by setting our
compass point at 𝐵. And we use the pencil point to
create arcs above and below the line segment. We can draw the whole circle if we
wish. But often, it’s easier to see the
diagram if we just draw two arcs. And that’s the arcs centered at 𝐵
done. And we need to do the same for arcs
centered at 𝐶.
So we take the compass, turn it
around. And this time, we place the sharp
pointed end at point 𝐶. Drawing two arcs above and below
the line segment will give us something like this. We notice that the pair of arcs
above and below the line segment have intersection points, which we can label as 𝑃
and 𝑄. By joining these points, we create
the line segment 𝑃𝑄, which is the perpendicular bisector of the line segment
𝐵𝐶. We are told that this line segment
bisects line segment 𝐵𝐶 at the point 𝐷, which we can label on the diagram.
And because we have accurately
constructed a bisector using the compass, we know that the line segments 𝐵𝐷 and
𝐶𝐷 are congruent. We also know that since this is a
perpendicular bisector, then the measures of the angles 𝐴𝐷𝐶 and 𝐴𝐷𝐵 are both
90 degrees. Therefore, we can give the answer
that the measure of the required angle 𝐴𝐷𝐶 is 90 degrees.
It’s worth noting that following
these steps will always allow us to construct the perpendicular bisector of a line
segment. And the more we practice these
steps, the easier they become to follow.
