Video Transcript
Which construction is illustrated
below? (A) A bisector of an angle. (B) A bisector of a line
segment. (C) A straight line parallel to
another line. (D) An angle congruent to another
angle. Or (E) a perpendicular from a point
lying outside a straight line.
In the diagram, we can observe that
we have the horizontal line 𝐴𝐵 and another line passing through 𝐶 which
intersects line 𝐴𝐵. We also have several arcs marked in
red on the figure, which are created during construction of this figure. Of the five given options, we need
to determine which type of construction this is. Of these five options, the most
likely one is that this is a perpendicular from a point lying outside a straight
line, where the straight line is 𝐴𝐵 and 𝐶 is such a point outside the line. However, let’s remind ourselves of
the steps that we would take if we wanted to construct a perpendicular of a straight
line.
Here, we have the line 𝐴𝐵 and a
point 𝐶 anywhere outside the line. It could be very close to the line
or very far away, just not on the line. If it was on the line, we would use
a different type of construction. And it’s worth noting that 𝐶
doesn’t need to be halfway along the line, even if in the original diagram it does
look about halfway along. 𝐶 can be anywhere. It’s also useful to leave a bit of
space below the diagram so that we have room for the arcs we need to draw, which
brings us to the first step.
We need to draw a circle which has
a center at point 𝐶 and which intersects the line 𝐴𝐵 twice. And just like with any
construction, we’ll need to use one of these tools: a compass. So we set the sharp end of the
compass onto point 𝐶 and draw the arc of a circle which intersects line 𝐴𝐵 twice,
something like this. We can draw the whole circle if we
wish. But this part of the circle
intersecting the line is all that we need. We can label the points of
intersection as 𝐷 and 𝐸.
The next step is to trace circles
at 𝐷 and 𝐸 that intersect. So, this time, we place our compass
point on 𝐸. And we can create the arc on the
other side of the line, although it doesn’t matter if it’s on the same side of the
line as point 𝐶. And we can draw an arc like
this. Now, we will lift the compass and
place the pointed end onto point 𝐷 to create another arc which has the same radius,
which creates an arc like this. We can label the point of
intersection of these two arcs as 𝐹. By joining the points 𝐶 and 𝐹, we
get the line 𝐶𝐹, which is perpendicular to line 𝐴𝐵. And we know that it passes through
the point 𝐶.
Following these steps will always
allow us to create the perpendicular to a line through a point which does not lie on
the line. Comparing this with the original
diagram, we can see the identical features, including the arcs, which allows us to
conclude that this is a perpendicular from a point lying outside a straight
line. This perpendicular construction is
the answer given in option (E). None of the other options would
give an equivalent diagram.
