Answer :
We have one triangle inside another, of which the vertices of the smaller triangle are the midpoints of the sides of the bigger triangle.
We know the lengths of the sides: XZ = 28, KL = 42 and YZ = 76.
We can construct similar triangles, like YLJ and YZK.
Knowing that L is the midpoint of YZ, we know that YL = (1/2)*YZ. Then, the sides of YLJ are half the length of their corresponding sides of YZK.
In the same way we can relate each corresponding side as:
• LK is half as long as XY.
,• LJ is half as long as XZ.
,• KJ is half as long as YZ
Then, we can start by calculating XY.
As this side is the double of LK, its length is:
[tex]XY=2\cdot LK=2\cdot42=84[/tex]We continue with JY.
JY is half segment of XY, so its length is 42.
Finally, JK is half the length of YZ, so its length is:
[tex]JK=\frac{1}{2}\cdot YZ=\frac{1}{2}\cdot76=38[/tex]Answer:
XY = 84
JY = 42
JK = 38