If z is the centroid of angle rst, rz= 42, st= 74, tw= 51, zy=23 and find each measure

Check the picture below.

now, keep in mind that, a median, comes out of the vertices, and cuts the opposite side in two equal halves, thus is called a median.

now, there are 3 vertices, there are 3 medians, now they all meet at the centroid.

when they meet, the centroid cuts each median in a 2:1 ratio, so one side is twice as long as the small one.

with that in mind, we know ST = 74, therefore XT is just one of the halves, 74/2.

we know TW = 51, and we know the centroid cuts it in a 2:1 ratio, so if we divide 51 by 3, 2 pieces go to TZ and 1 piece to ZW.  51/3 = 17, so TZ takes 2 of those pieces or 17+17.

well, you know what ZW gets by now anyway.

we know RZ = 42, and we also know because of the 2:1 ratio, RZ is twice as large as XZ, so XZ is half of RZ, or 42/2.

we know ZY = 23, and because of the 2:1 ratio, we know ZY is smaller piece so SZ is twice as large, so SZ is 23*2.  Therefore SY is just ZY + SZ.