Let ABCD be a parallelogram. Let M be the midpoint of AB and N be the midpoint of AD. Diagonal BD intersects CM and CN at P and Q respectively find PC/BD.
To find PC/BD in the given parallelogram ABCD, we need to use the properties of midpoints and parallel lines.
Step 1: Draw the parallelogram ABCD and mark the midpoints M and N on sides AB and AD, respectively.
Step 2: Draw the diagonal BD, which intersects CM at point P and CN at point Q.
Step 3: Notice that in a parallelogram, the diagonals bisect each other. So, BD also bisects CM and CN.
Step 4: Since M is the midpoint of AB, we know that AM = MB and BM is parallel to CD. Similarly, since N is the midpoint of AD, AN = ND and DN is parallel to BC.
Step 5: Apply the Midpoint Theorem, which states that in a line segment, the segment connecting the midpoints of two sides is parallel to the third side and half of its length. In this case, we can use the Midpoint Theorem in triangle ABD for sides AM, DN, and BD.
Step 6: Using the Midpoint Theorem, we can say that PC is parallel to AM. Similarly, QD is parallel to DN.
Step 7: Since PC is parallel to AM and BM is parallel to CD, triangles MBC and PCN are similar by AA (Angle-Angle) similarity.
Step 8: By the similarity of triangles MBC and PCN, we can conclude that the corresponding sides are proportional. So we have:
(PC / MC) = (BC / BM)
Step 9: Since BM = CD (opposite sides of a parallelogram are equal) and BC = CD (definition of a parallelogram), we have:
(PC / MC) = 1/2
Step 10: Since we know that BD bisects CM, we can say that MC = 2 * PC. Substitute this value into the equation:
(PC / (2 * PC)) = 1/2
1/2 = 1/2
Step 11: Therefore, PC / BD = 1/2.
So, in the given parallelogram ABCD, PC/BD is equal to 1/2.