Video Transcript
Use the figure to determine the
values of 𝑥 and 𝑦.
In the figure, we could observe
that there are four different lengths given on the figure. And these are all either in terms
of 𝑥 or 𝑦. We can also observe from the
markings on the figure that line segments 𝐸𝐷 and 𝐶𝐵 are parallel. And we have two congruent line
segments: 𝐴𝐷 and 𝐷𝐵. That means that the point 𝐷 can be
described as the midpoint of 𝐴𝐵. As we have a pair of parallel lines
and the midpoint of a line segment, we can apply one of the triangle midsegment
theorems.
The line segment passing through
the midpoint of one side of a triangle that is also parallel to another side of the
triangle bisects the third side of the triangle. 𝐸𝐷 is this line segment. It passes through the midpoint of a
side in the triangle, and it’s parallel to another side. Therefore, the third side of the
triangle, the line segment 𝐴𝐶, is bisected by line segment 𝐸𝐷. So, we can say that 𝐸 must be the
midpoint of this side. And importantly, this fact tells us
that the line segments 𝐴𝐸 and 𝐶𝐸 must be congruent.
Now, we were given the lengths of
these line segments. So we can fill them into the
equation. We have three 𝑥 minus seven equals
two 𝑥 plus one. We can then rearrange this so that
we have the terms with 𝑥 on one side and terms which are just numbers on the other
side. By subtracting two 𝑥 and adding
seven to both sides, either in one step or in two steps, we have that three 𝑥 minus
two 𝑥 equals one plus seven, which simplifies to 𝑥 equals eight.
And so we have found the first
required value of 𝑥. However, we’ll need some other
information to help us determine the value of 𝑦. This time, we can’t say that the
two line segments 𝐸𝐷 and 𝐶𝐵 are congruent. In fact, they don’t even look
congruent. So, we’ll need another of the
triangle midsegment theorems, this one. It states that the length of the
line segment joining the midpoints of two sides of a triangle is equal to half the
length of the third side.
We’ve already worked out that line
segment 𝐸𝐷 is a line segment joining the midpoints of two sides of a triangle. Therefore, its length is half that
of the third side, which is the line segment 𝐶𝐵. So, we can write that the length of
𝐸𝐷 equals one-half 𝐶𝐵. Or alternatively, by multiplying by
two, we can write that two times 𝐸𝐷 is 𝐶𝐵.
Now, we can substitute in the given
lengths from the diagram. This gives us two times four 𝑦
minus two equals nine 𝑦 minus seven. Next, we can multiply the two
across the parentheses to simplify the left-hand side, which gives us eight 𝑦 minus
four equals nine 𝑦 minus seven. As we saw before, we want to
combine the like terms. As the larger value of nine 𝑦 is
on the right-hand side, it may be easier to combine the 𝑦-terms on the right-hand
side. And so, by subtracting eight 𝑦 and
adding seven to both sides, we can collect the terms. Simplifying this gives three equals
𝑦 or 𝑦 equals three. And that’s both values of 𝑥 and 𝑦
calculated. They are eight and three,
respectively.
