Video Transcript
Given that 𝐴𝐵𝐶 is a
triangle, use a ruler and a pair of compasses to draw the triangle shown and
bisect angle 𝐶 and angle 𝐵 by the bisectors the ray from 𝐶 through 𝐷 and the
ray from 𝐵 through 𝐷 that intersect at 𝐷. Use the ruler to measure the
perimeter of the triangle 𝐷𝐵𝐶 to the nearest one decimal place. Option (A) 17.6 centimeters,
option (B) 17.9 centimeters, option (C) 16.9 centimeters, option (D) 18.4
centimeters. Or is it option (E) 19.1
centimeters?
In this question, we are told
to construct a triangle 𝐴𝐵𝐶 of given lengths and then construct the angle
bisectors of two of its angles. We know that these will
intersect at a point, and we can label this point 𝐷. We need to find this point and
then measure the perimeter of triangle 𝐷𝐵𝐶 to the nearest tenth of a
centimeter.
To begin, we might be tempted
to use the given diagram. However, there might be
inaccuracies in the diagram. Instead, we should construct
the triangle ourselves to minimize errors. To do this, we need to start
with sketching any side of the triangle. Let’s say we draw a line of
length nine centimeters as shown. We can then trace a circle of
radius six centimeters centered at 𝐶 and a circle of radius five centimeters
centered at 𝐵 and label the point of intersection 𝐴. If we connect the vertices with
sides as shown, then we have constructed the triangle with the given side
lengths.
We now need to bisect the
angles at 𝐶 and 𝐵. We can do this by recalling the
method for constructing an angle bisector using a pair of compasses and a
ruler. We can start by bisecting the
angle at 𝐶. We first need to trace a circle
centered at 𝐶 that intersects both sides of the angle. We will call these points 𝐸
and 𝐹 as shown. Next, we trace congruent
circles centered at 𝐸 and 𝐹 that intersect at a point on the same side as the
angle we want to bisect. We will call this point 𝐺 as
shown. Finally, we can conclude that
the line between 𝐶 and 𝐺 is the bisector of the angle. We will extend this bisector to
the opposite side of the triangle.
We now need to follow this same
process to bisect the angle at 𝐵. We start by tracing a circle at
𝐵 and labeling the points of intersection between the circle and the sides of
the angle 𝐼 and 𝐻 as shown. Next, we trace congruent
circles at 𝐼 and 𝐻 and label the point of intersection between the circles
𝐽. We can then conclude that the
line between 𝐽 and 𝐵 is the bisector of the angle at 𝐵. We can extend this line to the
opposite side of the triangle. We can now label the point of
intersection between the bisectors 𝐷.
We now want to estimate the
perimeter of triangle 𝐷𝐵𝐶, that is, the sum of the side lengths. If we measure the two unknown
side lengths of this triangle with a ruler, we can estimate that 𝐶𝐷 is
approximately 5.2 centimeters long and 𝐵𝐷 is approximately 4.2 centimeters
long. Adding these lengths together
gives us a perimeter of 18.4 centimeters to the nearest tenth of a centimeter,
which we can see agrees with option (D).
