In triangle 𝐾𝑀𝐻, 𝐾𝑄 equals two
and 𝑄𝑃 equals five 𝑥 minus seven. Find 𝑥.
First, we want to look at our
diagram and see what we know. We see that the points 𝐽, 𝑃, and
𝐿 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 𝐻𝐿, 𝑀𝐽, and 𝐾𝑃
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 𝐾𝑄 is two and that 𝑄𝑃 is five 𝑥 minus seven. Because point 𝑄 is the centroid,
𝐾𝑄 is equal to two-thirds of 𝐾𝑃. That is, the distance from the
vertex to the centroid is two-thirds of the distance of the median. And then 𝑄𝑃 equals one-third of
the distance of 𝐾𝑃.
And so we can say that 𝐾𝑄 is
equal to two times 𝑄𝑃. Or we can say that 𝑄𝑃 is equal to
one-half of 𝐾𝑄 because two-thirds is equal to one-third times two or one-third is
equal to two-thirds times one-half. Using the formula 𝑄𝑃 is equal to
one-half times 𝐾𝑄, we plug in the values we know for 𝑄𝑃 and 𝐾𝑄. And we get five 𝑥 minus seven
equals one-half times two. One-half times two is one, so we
have five 𝑥 minus seven equals one. And we add seven to both sides. Five 𝑥 equals eight, and 𝑥 will
equal eight divided by five, which is 1.6.
At this point, it’s probably worth
plugging 𝑥 back in to our five 𝑥 minus seven to make sure that this answer seems
reasonable. We know that 𝑄𝑃 will be equal to
five times 𝑥 minus seven. Five times 1.6 is eight; eight
minus seven equals 𝑄𝑃. And that means 𝑄𝑃 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 𝑥 equals 1.6.