In the given figure AO/OC = DO/OB = 1/2 and AD = 4cm. Then the value of BC is:

Key idea: place the figure on a coordinate plane so AD is horizontal and use the given ratios to find the intersection coordinates of the diagonals.

• Put A at (0,0) and D at (4,0) so AD = 4 cm along the x-axis. Let B and C lie on a horizontal line y = -H with B at (0,-H) and C at (L,-H). Then BC = L.

• Diagonal AC runs from (0,0) to (L,-H). Since AO:OC = 1:2, point O is one third of the way from A to C, so

  O = (L/3, -H/3).

• Diagonal DB runs from (4,0) to (0,-H). Since DO:OB = 1:2, point O is one third of the way from D to B, so

  O = (4 + 1/3*(0-4), 0 + 1/3*(-H-0)) = (4 - 4/3, -H/3) = (8/3, -H/3).

• Compare the x-coordinates of O from both diagonals: L/3 = 8/3, so L = 8. Therefore BC = 8 cm.

Answer: BC = 8 cm