a In the quadrilateral 1 given below AB DC E and F are the midpoints of AD and BD respectively Prove...

Question

a In the quadrilateral 1 given below AB DC E and F are the midpoints of AD and BD respectively Prove that i G is the midpoint of BC ii EG AB DC b In the quadrilateral 2 given below AB DC EG If E is midpoint of AD prove that i G is the midpoint of BC ii 2EG AB CD c In the quadrilateral 3 given below AB DC E and F are midpoint of nonparallel sides AD and BC respectively Calculate i EF if AB 6 cm and DC 4 cm ii AB if DC 8 cm and EF 9 cm

Accepted Answer

a It is given that AB DC E and F are midpoints of AD and B To prove i G is midpoint of BC ii EG AB DC Proof In ABD F is the midpoint of BD DF BF E is the midpoint of AD EF AB and EF AB 1 It is given that AB CD EG CD F is the midpoint of BD FG DC G is the midpoint of BC FG DC 2 By adding both the equations EF FG AB DC Taking as common EG AB DC Therefore it is proved b It is given that Quadrilateral ABCD in which AB DC EG E is the midpoint of AD To prove i G is the midpoint of BC ii 2EG AB CD Proof AB DC EG AB So we get EG DC In DAB E is the midpoint of BD and EF AB 1 In BCD F is the midpoint of BD and FG DC FG CD 2 By adding both the equations EF FG AB CD Taking out the common terms EG AB CD Hence it is proved c It is given that A quadrilateral in which AB DC E and F are the midpoints of nonparallel sides AD and BC To prove i EF if AB 6 cm and DC 4 cm ii AB if DC 8 cm and EF 9 cm Proof We know that The length of line segment joining the midpoints of two nonparallel sides is half the sum of the lengths of the parallel sides E and F are the midpoints of AD and BC EF AB CD 1 i AB 6 cm and DC 4 cm Substituting in equation 1 EF 6 4 By further calculation EF 10 5 cm ii DC 8 cm and EF 9 cm Substituting in equation 1 EF AB DC By further calculation 9 AB 8 18 AB 8 So we get 18 8 AB AB 10 cm

ML Aggarwal Solutions Class 9 Mathematics Solutions for Mid Point Theorem Exercise 11 in Chapter 11 - Mid Point Theorem