Video Transcript
Draw a triangle 𝐴𝐵𝐶, where 𝐴𝐵
equals two centimeters, 𝐵𝐶 equals five centimeters, and 𝐴𝐶 equals six
centimeters. Draw the perpendicular bisectors of
line segments 𝐴𝐵, 𝐵𝐶, and 𝐴𝐶. What do you observe about the three
perpendicular bisectors? (A) They all intersect at one
point. (B) They are all equal in
length. (C) They are all medians of the
triangle.
There are two steps to carry out in
this question: firstly to draw a triangle and then to draw the perpendicular
bisectors of the sides. So, let’s begin by drawing, or
rather constructing, the triangle. And we need to pay careful
attention to the lengths that these sides need to be. We have sides of two centimeters,
five centimeters, and six centimeters. Now, it doesn’t matter which we
draw first. But it can sometimes be helpful to
draw the longest side on the base of the triangle.
So here we have measured side 𝐴𝐶
with a ruler to make it six centimeters long. We now need to draw sides of two
centimeters and five centimeters. However, if we tried to draw these
sides by just using a ruler, it could take quite a long time before we successfully
get these line segments to meet at a single point and be the correct length. Therefore, the best tool that we
can use is a compass. We use a compass like this for lots
of different types of construction, as we’ll see with the line segment bisector
shortly. But first, let’s see how we can use
it to draw the line segment 𝐵𝐶 of length five centimeters.
We move our compass so that the
sharp end is in point 𝐶, because this is line segment 𝐵𝐶. So, it needs to pass between 𝐶 and
𝐵. We should use a ruler to make sure
that the distance between the compass point and the sharp tip of the pencil is
exactly five centimeters. So, when we draw the arc of the
circle, this will represent a range of the possible points that 𝐵 could lie on.
We are now going to do the same for
the line segment 𝐴𝐵. So, we turn the compass and place
the sharp point on point 𝐴. And this time, we make sure that
the distance between the point and the pencil tip is set to two centimeters. Drawing the second arc would create
something like this.
So now we know that at the point of
intersection of these two arcs, the length of 𝐵𝐶 is five centimeters and the
length of 𝐴𝐵 is two centimeters. And we have accurately constructed
triangle 𝐴𝐵𝐶.
Next, we need to draw the
perpendicular bisectors of the three sides of the triangle. And although the word used here is
“draw,” we should use a construction method with a compass. We can recall that there are
certain steps to this process. If we have a line segment 𝐴𝐶 that
we need to bisect, then we first need to set the radius of the compass to be greater
than one-half the length of 𝐴𝐶. Then, we trace two circles of this
radius centered at 𝐴 and 𝐶. Let’s see what that would look like
on the diagram.
We place the compass point at point
𝐴. And we draw arcs of this radius
above and below the line segment 𝐴𝐶. Now, without changing the size of
the compass, we place the compass on point 𝐶 to draw the arcs of a circle centered
at point 𝐶, which gives us something like this. Notice that the pairs of arcs above
and below the line segment intersect at two points, which we can label 𝐷 and
𝐸. By joining the two points 𝐷 and
𝐸, we have created the line segment 𝐷𝐸, which is the perpendicular bisector of
line segment 𝐴𝐶.
Now, we need to find the
perpendicular bisectors of the remaining two sides. And we can take side 𝐵𝐶 next. Often, when we are constructing the
perpendicular bisector of a line segment which isn’t a horizontal line, we may find
it easier to turn the page around so that the line is horizontal to us. We can also adjust the terminology
of our method instructions so that we are bisecting line segment 𝐵𝐶 to help keep
us right. Placing our compass point on 𝐵 and
making sure that the length of the compass is set to be greater than half the length
of line segment 𝐵𝐶, we can draw arcs above and below the line segment like
this. Next, we place our compass point at
𝐶 and create arcs above and below the line segment.
As we saw before, defining the
points of intersection of the arcs as 𝐹 and 𝐺 and joining them, we know that line
segment 𝐹𝐺 is the perpendicular bisector of line segment 𝐵𝐶. And now we can complete the final
perpendicular bisector for the side 𝐴𝐵. We create arcs centered at 𝐴 and
𝐵. And defining the points of
intersection as 𝐻 and 𝐾, we have created the perpendicular bisector 𝐻𝐾 of line
segment 𝐴𝐵.
We can now observe that the three
perpendicular bisectors meet at a single point, which is in fact a general result
for all triangles. The point of intersection of the
perpendicular bisectors of the sides of a triangle will be inside the triangle for
acute triangles, outside the triangle in obtuse triangles, or on the hypotenuse of
right triangles.
Therefore, by accurate
construction, we have proved the result that the perpendicular bisectors of a
triangle meet at one point, which was the answer given in option (A).
