Video Transcript
Consider the shown triangle
π΄π΅πΆ. For each of the following
statements, fill in the blank with is greater than, is less than, or is equal
to. π΄πΆ what π΄π΅, π΄π΅ what π΅π·,
πΆπ· what π΄π·, and π΅π· what π΄πΆ.
In this question, we are given a
figure of a triangle. And we want to use this figure to
compare the lengths of various line segments in the figure. To do this, letβs start by
highlighting the first pair of line segments whose lengths we want to compare: π΄πΆ
and π΄π΅. We can then note that these are
both sides in triangle π΄π΅πΆ. So we can compare the lengths of
the sides by recalling the side comparison theorem in triangles. This tells us that if one side of a
triangle is opposite an angle of larger measure, then it must be the longer of the
two sides. More formally, this tells us that
if we have a triangle πππ and the measure of angle π is greater than the measure
of angle π, then ππ is longer than ππ since it is opposite an angle of larger
measure in the triangle.
Therefore, we can compare the
lengths of these two sides by comparing the measures of the angles opposite them in
triangle π΄π΅πΆ. We see that π΄πΆ is opposite the
right angle of measure 90 degrees and π΄π΅ is opposite an angle of measure 30
degrees. Hence, since π΄πΆ is opposite an
angle of larger measure than π΄π΅, it must be the longer side. So π΄πΆ is greater than π΄π΅.
It is worth noting that the angle
comparison theorem tells us that the side opposite the right angle in any right
triangle is always the longest, since the right angle is the angle with the largest
measure. We call this side the
hypotenuse. We can then use this idea to answer
the fourth part of this question. π΄πΆ is the longest side in
triangle π΄π΅πΆ, so it is longer than π΅πΆ, which in turn is longer than π΅π·. So, we know that π΅π· is less than
π΄πΆ.
Letβs now try to apply the side
comparison theorem to the sides in the second part of the question. First, we highlight the two sides
whose lengths we want to compare as shown. We see that these are both side
lengths in triangle π΄π΅π·. We can use the side comparison
theorem in triangles to compare the lengths of these sides by comparing the measure
of the angles opposite the sides in triangle π΄π΅π·. We see that angle π΅π΄π· has
measure 30 degrees. And we can find the measure of
angle π΄π·π΅ by recalling that the sum of the internal angle measures in a triangle
is 180 degrees. Therefore, the measure of angle
π΄π·π΅ is equal to 180 degrees minus 90 degrees minus 30 degrees, which is equal to
60 degrees. We can then see that π΄π΅ is
opposite an angle of larger measure, so it must be the longer of the two sides. Hence, π΄π΅ is greater than
π΅π·.
Letβs apply this process one final
time to the pair of sides in the third part of this question. We can start by highlighting the
two sides whose lengths we want to compare on the figure. We see that these are both side
lengths in triangle π΄πΆπ·. If we look at the angle measures
opposite the sides in triangle π΄πΆπ·, we see that both sides are opposite angles of
measure 30 degrees. Since the sides are opposite angles
of equal measure, we can use the isosceles triangle theorem to conclude that the
sides have the same length. So, πΆπ· is equal to π΄π·.