a In the quadrilateral 1 given below AD BC P Q R and S are midpoints of AB BD CD and AC respectively...

Question

a In the quadrilateral 1 given below AD BC P Q R and S are midpoints of AB BD CD and AC respectively Prove that PQRS is a rhombus b In the figure 2 given below ABCD is a kite in which BC CD AB AD E F G are midpoints of CD BC and AB respectively Prove that i EFG ii The line drawn through G and parallel to FE bisects DA

Accepted Answer

a It is given that A quadrilateral ABCD in which AD C P Q R and S are midpoints of AB BD CD and AC To prove PQRS is a rhombus Proof In ABD P and Q are mid points of AB and BD PQ AD and PQ AB 1 In BCD R and Q are mid points of DC and BD RQ BC and RQ BC 2 P and S are midpoints of AB and AC PS BC and PS BC 3 AD BC Using all the equations PS RQ and PQ PS RQ Here PS RQ and PS RQ PQRS is a parallelogram PQ RS PS RQ PQRS is a rhombus Therefore it is proved ii It is given that ABCD is a kite in which BC CD AB AD E F G are midpoints of CD BC and AB To prove i EFG ii The line drawn through G and parallel to FE bisects DA Construction Join AC and BD Construct GH through G parallel to FE Proof i We know that Diagonals of a kite interest at right angles MON 1 In BCD E and F are midpoints of CD and BC EF DB and EF DB 2 EF DB MF ON Here MON MFN MFN By further calculation MFN 180 90 Hence it is proved ii In ABD G is the midpoint of AB and HG DB Using equation 2 EF DB and EF HG HG DB Here H is the midpoint of DA Therefore the line drawn through G and parallel to FE bisects DA

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