Video Transcript
From the figure below, determine
the correct inequality from the following: 𝐴𝐵 is greater than 𝐶𝐵, 𝐴𝐵 is less
than 𝐶𝐵, 𝐴𝐵 is greater than 𝐴𝐶, or 𝐴𝐶 is less than 𝐶𝐵.
Looking at the diagram, we can see
that 𝐴𝐵, 𝐶𝐵, and 𝐴𝐶 all represent the lengths of sides of a triangle. We’ve been given four possibilities
for relationships that could exist between the lengths of different pairs of
sides. We haven’t been given any lengths
in the diagram. Instead, we’ve been given some
information about some of the angles. This suggests that we need to
consider the relationship between the lengths of sides and the size of angles in a
triangle. And therefore, we’re going to
approach this question using the angle–side triangle inequality.
Here’s what the angle–side triangle
inequality tells us. If one angle of a triangle has a
greater measure than another angle, then the side opposite the greater angle is
longer than the side opposite the lesser angle. Basically, what this means is that
the longest side of a triangle is opposite the largest angle. The shortest side is opposite the
smallest angle. And the middle side is opposite the
middle angle. In the diagram, however, we’ve only
currently got the size of one of the angles in the triangle. So we need to consider how we can
find the other angles.
First of all, let’s consider angle
𝐴𝐵𝐶. We can see that the lines 𝐴𝐷 and
𝐶𝐵 are parallel as they’ve been marked with blue arrows on their lengths. The line 𝐴𝐵 is a transversal
through these parallel lines. And therefore, we can see that the
angle 𝐴𝐵𝐶 and the angle of 66 degrees are alternate interior angles. Which means that they’re
congruent. So angle 𝐴𝐵𝐶 is also 66
degrees.
Now that we know the measures of
two of the angles in the triangle, we can calculate the third because the angle sum
in a triangle is always 180 degrees. So angle 𝐴𝐶𝐵 can be found by
subtracting 52 degrees and 66 degrees from 180 degrees. It’s 62 degrees.
So now that we know the sizes of
all three angles in the triangle, we can deduce something about the lengths of the
three sides. The largest angle in the triangle
is 66 degrees. And the angle–side triangle
inequality tells us that the longest side of the triangle will be opposite this
angle. So the longest side of the triangle
is the side 𝐴𝐶. The second biggest angle in the
triangle is the angle of 62 degrees which is opposite the side 𝐴𝐵. This means then that 𝐴𝐵 is the
second longest side of the triangle. The smallest angle of 52 degrees is
opposite the shortest side of the triangle. So 𝐶𝐵 is the shortest side.
Now that we have the three sides of
the triangle ordered from longest to shortest, we can turn our attention to the four
inequalities and determining which are true. Firstly, is 𝐴𝐵 greater than
𝐶𝐵? Yes, 𝐴𝐵 appears above 𝐶𝐵 in the
list. Which means this first inequality
is true. Is 𝐴𝐵 less than 𝐶𝐵? Well, this is the reverse of the
inequality that we’ve just shown to be true. Therefore, this one must be
false. Thirdly, is 𝐴𝐵 greater than
𝐴𝐶? No, 𝐴𝐶 is the longest side of the
triangle. So this inequality is also
false. And finally, is 𝐴𝐶 less than
𝐶𝐵? Again, this is false. 𝐴𝐶 is the longest side of the
triangle.
So we can conclude that of the four
inequalities, only one is true. 𝐴𝐵 is greater than 𝐶𝐵.
