From the Centroid Theorem find three equalities true for â–ł KHM, then use each of them to calculate x, y, and z.
x=4.75, y=6, z=1
Practice makes perfect
We are told that J, P, and L are the midpoints of KH, HM, and MK. Let's mark this information on the diagram.
Let's review that a median of a triangle is a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side. Thus, MJ, KP, and HL are the medians of the triangle â–ł KHM. Now we can use the Centroid Theorem.
The medians of a triangle inersect
at a point called the centroid that is
two thirds of the distance from each vertex
to the midpoint of the opposite side.
Medians MJ, KP, and HL intersect at the point Q, so Q is the centroid of â–ł KHM. By the means of the above theorem, we can conclude the following.
KQ= 23KP
HQ= 23HL
MQ= 23MJ
Now we can use these equalities to find the values of x, y, and z.
Finding x
In order to find x, let's use the first equality.
KQ= 23KP
From the diagram, we know that the measure of KQ is 7 and the measure of QP is 2x-6. These segments form KP. Adding their measures by the Segment Addition Postulate, we can find the measure of KP.
Let's now use the second equality.
HQ= 23HL
From the diagram, we know that HQ measures y and QL measures 3. Since these segments add up to HL, we can find its measure by adding this segment's measures.
HL=y+ 3
Now that we know both HQ and HL, let's substitute the corresponding values into the above equality.
Finally, we can use the third equality to find the value of z.
MQ= 23MJ
We are given on the diagram that MQ measures 4 and QJ measures 2z. Adding these values, we can find the measure of MJ.
MJ=2z+4
Let's now substitute 4 for MQ and 2z+4 for MJ into our equality, and solve it for z.
