2. A diagonal of a parallelogram divides it into two congruent triangles.
3. In a parallelogram,
(i) opposite sides are equal (ii) opposite angles are equal(iii) diagonals bisect each other
4. A quadrilateral is a parallelogram, if
(i) opposite sides are equal or (ii) opposite angles are equal or (iii) diagonals bisect each other or (iv) a pair of opposite sides is equal and parallel
5. Diagonals of a rectangle bisect each other and are equal and vice-versa.
6. Diagonals of a rhombus bisect each other at right angles and vice-versa.
7. Diagonals of a square bisect each other at right angles and are equal, and vice-versa.
8. The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.
9. A line through the mid-point of a side of a triangle parallel to another side bisects the third side.
10. The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.
Answer: In the given quadrilateral ABCD diagonals AC and BD bisect each other at right angle. We have to prove that AB=BC=CD=AD
In DO=OB (O is the midpoint)AO=AO (common side)
(Right angle)So, So, AB=ADSimilarly AB=BC=CD=AD can be proved which means that ABCD is a rhombus.
4. Show that the diagonals of a square are equal and bisect each other at right an-gles.
Answer: In the figure given above let us assume that So, Hence, DO=AO (Sides opposite equal angles are equal)Similarly AO=OB=OC can be provedThis gives the proof of diagonals of square being equal.
5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
Answer: Using the same figure,If DO=AOThen (Angles opposite to equal sides are equal)So, all angles of the quadrilateral are right angles making it a square.
6. Diagonal AC of a parallelogram ABCD bisects A . Show that(i) it bisects C also,
Answer: ABCD is a parallelogram where diagonal AC bisects
In (diagonal is bisecting the angle)
AC=AC (Common side)AD=BC (parallel sides are equal in a parallelogram)Hence, So, This proves that AC bisects as well Now let us assume another diagonal DB intersecting AC on O.As it is a parallelogram so DB will bisect AC and vice versaIn
(opposite angles are equal in parallelogram so their halves will be equal)AO=CODO=DOHence, So, As diagonals are intersecting at right angles so it is a rhombus
7. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that:(i) ∆ APD ∆ CQB(ii) AP = CQ(iii) ∆ AQB ∆ CPD(iv) AQ = CP(v) APCQ is a parallelogram
Answer: In DP=BQ (given)AD=BC (opposite sides are equal)
(opposite angles’ halves are equal)Hence, So, AP=CQ Proved
In AB=CD (Opposite sides are equal)DP=BQ (given)
( opposite angles’ halves are equal)Hence, So, AQ=CP Proved
(corresponding angles of congruent triangles APD & CQB)In
(from previous proof)DP=BQ (given)PQ=PQ (common Side)So, So, With equal opposite angles and equal opposite sides it is proved that APCQ is a paral-lelogram
8. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD. Show that(i) ∆ APB ∆ CQD(ii) AP = CQ
A B
C D P
Q
Answer: In (alternate angles of transversal DB)
AB=CD (right angles)
Hence, So, AP=CQ
9. In ∆ ABC and ∆ DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively. Show that
A
B C
D
E F
(i) quadrilateral ABED is a parallelogram(ii) quadrilateral BEFC is a parallelogram(iii) AD || CF and AD = CF(iv) quadrilateral ACFD is a parallelogram(v) AC = DF
In quadrilateral ABEDAB= EDAB||EDSo, ABED is a parallelogram (opposite sides are equal and parallel)So, BE||AD ------------ (1)Similarly quadrilateral ACFD can be proven to be a parallelogramSo, BE||CF ------------ (2)From equations (1) & (2)It is proved that AD||CFSo, AD=CFSimilarly AC=DF and AC||DF can be proved
10. ABCD is a trapezium in which AB || CD and AD = BC. Show that(i) A = B(ii) C = D(iii) ∆ ABC ∆ BAD(iv) diagonal AC = diagonal BD[Hint : Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]
A B E
C D
Answer: In EC=AD (Opposite sides are equal in parallelogram)AD=BC (given)So, BC=EC
Similarly AC||PQ can be proven which will prove that PQRS is a parallelogram.
2. ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.
A
B
C
D
P Q
R S
Answer: Following the method used in the previous question it can be proved that PQRS is a parallelogram. To prove it to be a rectangle we need to prove that
In DS=CR=BQ=AP=DR=CQ=BP=AS (All sides of rhombus are equal and PQRS are mid-points)
So, So, Hence, Or, As,
Similarly Hence, PQRS is a rectangle.3. ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F. Show that F is the mid-point of BC.
Answer: In DG=GBA parallel line to the base originating from mid point of second side will intersect at the midpoint of the third side.AB||DCAB||EFSo, EF||DCSo, In EG||ABE is the mid point of ADSo, G is the mid point of DBNow, in GF||DCG is the mid point of BDSo, F will be mid point of BC ( Mid point theorem)
4. In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respec-tively. Show that the line segments AF and EC trisect the diagonal BD.
Answer: In AD=BC (Opposite sides are equal in parallelogram)BF=DE (Half of opposite sides of parallelogram)
(Opposite angles are equal)So, Hence, AE=CFIn quadrilateral AECFEC||AF & EC=AFAE=CFSo, AE||CFSo, AECF is a parallelogram.
In PE||QC (proved earlier by proving AE||CF)E is the mid point of DCSo, P is the mid point of DQSo, DP=PQ
FQ||APF is the mid point of ABSo, PQ=QBSo, DP=PQ=QB proved
5. Show that the line segments joining the mid-points of the opposite sides of aquadrilateral bisect each other.
Answer: ABCD is a quadrilateral in which P, Q, R, & S are mid points of AB, BC, CD & ADIn SR is touching mid points of CD and ADSo, SR||ACSimilarly following can be provedPQ||ACQR||BDPS||BDSo, PQRS is a parallelogram.PR and QS are diagonals of the parallelogram PQRS, so they will bisect each other.
6. ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that(i) D is the mid-point of AC (ii) MD AC
(iii) CM = MA =
A
B
C
M
D
Answer: DM||BCM is the mid point of AB So, D is the mid point of AC (Mid point theorem)
