Video Transcript
In triangle πΎππ», πΎπ equals two
and ππ equals five π₯ minus seven. Find π₯.
First, we want to look at our
diagram and see what we know. We see that the points π½, π, and
πΏ divide each side of the triangles in half. And then we have lines from each of
the vertices to those points. The distance from a vertex to a
midpoint is the median. And that means π»πΏ, ππ½, and πΎπ
are all medians of this triangle. And we know the point of
concurrency for three medians is the centroid. We can also write down some other
information we know, that πΎπ is two and that ππ is five π₯ minus seven. Because point π is the centroid,
πΎπ is equal to two-thirds of πΎπ. That is, the distance from the
vertex to the centroid is two-thirds of the distance of the median. And then ππ equals one-third of
the distance of πΎπ.
And so we can say that πΎπ is
equal to two times ππ. Or we can say that ππ is equal to
one-half of πΎπ because two-thirds is equal to one-third times two or one-third is
equal to two-thirds times one-half. Using the formula ππ is equal to
one-half times πΎπ, we plug in the values we know for ππ and πΎπ. And we get five π₯ minus seven
equals one-half times two. One-half times two is one, so we
have five π₯ minus seven equals one. And we add seven to both sides. Five π₯ equals eight, and π₯ will
equal eight divided by five, which is 1.6.
At this point, itβs probably worth
plugging π₯ back in to our five π₯ minus seven to make sure that this answer seems
reasonable. We know that ππ will be equal to
five times π₯ minus seven. Five times 1.6 is eight; eight
minus seven equals ππ. And that means ππ is equal to
one. It is true that one is half of two
and that two is one times two. This confirms the proportionality
of our median and confirms that π₯ equals 1.6.
