The endpoints of one diagonal of rhombus are (0,-8) and (8,-4). If the coordinates of the third vertex are (1,0). what are the coordinates of the forth vertex?

Question

Evelyn Arboleda · Accepted Answer

The coordinates are #=(7,-12)# Let #A=(0,-8)# #C=(8,-4)# The midpoint of #AC# is #E=((0+8)/2, (-8-4)/2)=(4,-6)# Let #B=(1,0)# and #D=(x,y)# The mid point of #BD# is #E'=((x+1)/2,(y/2))# #E and E'# is the same point #(4,-6)=((x+1)/2,(y/2))# #=># #(x+1)/2=4# #x+1=8# #x=7# #y=-12# The point #D=(7,-12)#

Violet Aldrich · Answer

# (7,-12),# is the reqd. fourth vertex.

We know that a Rhombus is a Parallelogram, and, the Diagonals of a parallelogram bisect each other. This means that both the diagonals must have the same mid-point. Hence, the midpt. of the diagonal through vertices #(0,-8) and (8,-4)# is the same as that through the fourth vertex, say, #(x,y) and (1,0).# #:. ((0+8)/2,(-8-4)/2)=((x+1)/2,(y+0)/2).# # :. 8=x+1, -12=y.# # rArr (x,y)=(7,-12),# is the reqd. fourth vertex. Enjoy Maths.!

Parker Allen · Answer

undefined The coordinates of the fourth vertex of the rhombus can be found using the midpoint formula for a line segment. Given that the third vertex is (1,0) and one endpoint of the diagonal is (0,-8), we can find the midpoint between these two points, which will be the center of the rhombus. Then, using the midpoint and the other endpoint of the diagonal (8,-4), we can find the equation of the line passing through these two points. Finally, we can find the fourth vertex by finding the point on this line that is equidistant from the midpoint as the endpoint (8,-4). Solving these calculations yields the coordinates of the fourth vertex, which are (9,-12).