### Video Transcript

Given that π΄π΅πΆ is a
triangle, use a ruler and a pair of compasses to draw the triangle and bisect
angle πΆ by the bisector the ray from πΆ through π· that intersects the ray from
π΄ through π΅ at π·. Use the ruler to measure the
length of the line segment π΄π· to the nearest one decimal place. Option (A) 4.2 centimeters,
option (B) 3.2 centimeters, option (C) 4.7 centimeters, option (D) 3.7
centimeters. Or is it option (E) 5.2
centimeters?

In this question, we are asked
to draw a triangle of given lengths using a ruler and a pair of compasses,
bisect the angle at πΆ in a way to find a point π·, and then measure the length
of the line segment π΄π· to the nearest tenth of a centimeter.

Letβs start with the first part
of the question. We are asked to draw the
triangle with the given lengths. We might be tempted to skip
this step and use the given diagram. However, these are not always
accurate. So it is a good idea to
construct the triangle ourselves, although in this case the diagram is
accurate.

To construct a triangle with a
compass and ruler, we need to start by constructing one side of the
triangle. Letβs say we start with π΅πΆ,
which is of length five centimeters. We can then trace a circle of
radius seven centimeters centered at πΆ and a circle of radius eight centimeters
centered at π΅ and label either point of intersection between the circles
π΄. We can then use the radius of
each circle to note that triangle π΄π΅πΆ has the desired side lengths.

The second part of this
question asks us to bisect the angle at πΆ, that is, the internal angle of the
triangle. We can recall that the bisector
will split this angle into two congruent angles. We can also note that this
bisector will lie inside the triangle, so point π· will lie on the line segment
π΄π΅. We do not need to extend this
side. We then recall that we can
bisect angles by a construction. First, we trace a circle
centered at the vertex of the angle we want to bisect and label the points of
intersection between the circle and the sides π΄ prime, π΅ prime as shown. We can then sketch congruent
circles centered at π΄ prime and π΅ prime that intersect at a point on the same
side as the angle we want to bisect. We will call this point of
intersection πΈ.

We can then conclude that the
line between πΆ and πΈ bisects the angle at πΆ. We can extend this bisector to
intersect the opposite side and label the point of intersection π· as shown. We can then measure the length
of the line segment π΄π· with our ruler. And if we do, we get 4.7
centimeters to the nearest tenth of a centimeter, which is option (C).