1. The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
1. The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
Explanation:
Let x be the common ratio between the angles.
The sum of the interior angles of the quadrilateral = 360°
Hence,
3x+5x+9x+13x = 360°
⇒ 30x = 360°
⇒ x = 12°
Therefore, the angles of the quadrilateral are:
3x = 3×12° = 36°
5x = 5×12° = 60°
9x = 9×12° = 108°
13x = 13×12° = 156°
2. If the diagonals of a parallelogram are equal, then show that it is a rectangle.
2. If the diagonals of a parallelogram are equal, then show that it is a rectangle.
Explanation:
Given,
AC = BD
To show ABCD is rectangle if the diagonals are equal or a parallelogram.
To show ABCD is rectangle, one of its interior angles should be right-angled.
Therefore,
In ΔABC and ΔBAD,
AB = BA (Common)
BC = AD (Opposite sides of a parallelogram are equal)
AC = BD (Given)
Therefore, ΔABC ≅ ΔBAD [SSS congruency]
∠A = ∠B [Corresponding parts of Congruent Triangles]
also,
∠A+∠B = 180° (Sum of the angles on the same side of the transversal)
⇒ 2∠A = 180°
⇒ ∠A = 90° = ∠B
Therefore, ABCD is a rectangle.
3. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
3. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Explanation:
Let ABCD be quadrilateral of which diagonals bisect each other at right angles.
Given,
OA = OC
OB = OD
and ∠AOB = ∠BOC = ∠OCD = ∠ODA = 90°
To show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus, we have to prove that ABCD is parallelogram and AB = BC = CD = AD
Therefore,
In ΔAOB and ΔCOB,
OA = OC (Given)
∠AOB = ∠COB (Opposite sides of a parallelogram are equal)
OB = OB (Common)
Hence, ΔAOB ≅ ΔCOB [SAS congruency]
Thus, AB = BC [CPCT]
Similarly, we can prove,
BC = CD
CD = AD
AD = AB
, AB = BC = CD = AD
Opposite sides of a quadrilateral are equal. Therefore it is a parallelogram.
Hence, ABCD is rhombus as it is a parallelogram whose diagonals intersect at right angle.
4. Show that the diagonals of a square are equal and bisect each other at right angles.
4. Show that the diagonals of a square are equal and bisect each other at right angles.
Explanation:
Let ABCD be square and its diagonals AC and BD intersect each other at point O.
To prove that,
AC = BD
AO = OC
and ∠AOB = 90°
Proof,
In ΔABC and ΔBAD,
AB = BA (Common)
∠ABC = ∠BAD = 90°
BC = AD (Given)
ΔABC ≅ ΔBAD [SAS congruency]
Therefore,
AC = BD [CPCT]
diagonals are equal.
Now,
In ΔAOB and ΔCOD,
∠BAO = ∠DCO (Alternate interior angles)
∠AOB = ∠COD (Vertically opposite)
AB = CD (Given)
ΔAOB ≅ ΔCOD [AAS congruency]
Therefore,
AO = CO [CPCT].
Diagonal bisect each other.
Now,
In ΔAOB and ΔCOB,
OB = OB (Given)
AO = CO (diagonals are bisected)
AB = CB (Sides of the square)
ΔAOB ≅ ΔCOB [SSS congruency]
Also, ∠AOB = ∠COB
∠AOB+∠COB = 180° (Linear pair)
Therefore, ∠AOB = ∠COB = 90°
Diagonals bisect each other at right angles
5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
Explanation:
Given,
Let ABCD be a quadrilateral and its diagonals AC and BD bisect each other at right angle at O.
To prove that,
Quadrilateral ABCD is a square.
Proof,
In ΔAOB and ΔCOD,
AO = CO (Diagonals bisect each other)
∠AOB = ∠COD (Vertically opposite)
OB = OD (Diagonals bisect each other)
ΔAOB ≅ ΔCOD [SAS congruency]
Therefore,
AB = CD [CPCT] — (i)
Also,
∠OAB = ∠OCD (Alternate interior angles)
⇒ AB || CD
Now,
In ΔAOD and ΔCOD,
AO = CO (Diagonals bisect each other)
∠AOD = ∠COD (Vertically opposite)
OD = OD (Common)
ΔAOD ≅ ΔCOD [SAS congruency]
Therefore,
AD = CD [CPCT] — (ii)
Also,
AD = BC and AD = CD
⇒ AD = BC = CD = AB — (ii)
Also, ∠ADC = ∠BCD [CPCT]
and ∠ADC+∠BCD = 180° (co-interior angles)
⇒ 2∠ADC = 180°
⇒∠ADC = 90° — (iii)
One of the interior angles is a right angle.
Therefore, from (i), (ii) and (iii), given quadrilateral ABCD is a square.
6. Diagonal AC of a parallelogram ABCD bisects ∠A (see Fig. 8.19). Show that
(i) it bisects ∠C also,
(ii) ABCD is a rhombus.
6. Diagonal AC of a parallelogram ABCD bisects ∠A (see Fig. 8.19). Show that
(i) it bisects ∠C also,
(ii) ABCD is a rhombus.
Explanation:
(i) In ΔADC and ΔCBA,
AD = CB (Opposite sides of parallelogram)
DC = BA (Opposite sides of parallelogram)
AC = CA (Common Side)
, ΔADC ≅ ΔCBA [SSS congruency]
Therefore,
∠ACD = ∠CAB by CPCT
and ∠CAB = ∠CAD (Given)
⇒ ∠ACD = ∠BCA
Therefore,
AC bisects ∠C
(ii) ∠ACD = ∠CAD (Proved above)
⇒ AD = CD (Opposite sides of equal angles of a triangle are equal)
Also, AB = BC = CD = DA (Opposite sides of a parallelogram)
Therefore,
ABCD is a rhombus.
7. ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.
7. ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.
Explanation:
Given ,
ABCD is a rhombus.
AC and BD are its diagonals.
Hence ,
AD = CD (Sides of a rhombus)
∠DAC = ∠DCA (Angles opposite of equal sides of a triangle are equal.)
Also, AB || CD
⇒∠DAC = ∠BCA (Alternate interior angles)
⇒∠DCA = ∠BCA
, AC bisects ∠C.
Similarly,
Diagonal AC bisects ∠A.
Following the same method,
We can prove that the diagonal BD bisects ∠B and ∠D.
8. ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that:
(i) ABCD is a square
(ii) Diagonal BD bisects ∠B as well as ∠D.
8. ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that:
(i) ABCD is a square
(ii) Diagonal BD bisects ∠B as well as ∠D.
Explanation:
(i) ∠DAC = ∠DCA (AC bisects ∠A as well as ∠C)
⇒ AD = CD (Sides opposite to equal angles of a triangle are equal)
Also, CD = AB (Opposite sides of a rectangle)
AB = BC = CD = AD
Hence, ABCD is a square.
(ii) In ΔBCD,
BC = CD
⇒ ∠CDB = ∠CBD (Angles opposite to equal sides are equal)
Also, ∠CDB = ∠ABD (Alternate interior angles)
⇒ ∠CBD = ∠ABD
Therefore, bisects ∠B
Now,
∠CBD = ∠ADB
⇒ ∠CDB = ∠ADB
Therefore, BD bisects ∠B as well as ∠D.
9. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:
(i) ΔAPD ≅ ΔCQB
(ii) AP = CQ
(iii) ΔAQB ≅ ΔCPD
(iv) AQ = CP
(v) APCQ is a parallelogram
9. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:
(i) ΔAPD ≅ ΔCQB
(ii) AP = CQ
(iii) ΔAQB ≅ ΔCPD
(iv) AQ = CP
(v) APCQ is a parallelogram
Explanation:
(i) In ΔAPD and ΔCQB,
DP = BQ (Given)
∠ADP = ∠CBQ (Alternate interior angles)
AD = BC (Opposite sides of a parallelogram)
Therefore, ΔAPD ≅ ΔCQB [SAS congruency]
(ii) AP = CQ by CPCT because ΔAPD ≅ ΔCQB.
(iii) In ΔAQB and ΔCPD,
BQ = DP (Given)
∠ABQ = ∠CDP (Alternate interior angles)
AB = CD (Opposite sides of a parallelogram)
Therefore, ΔAQB ≅ ΔCPD [SAS congruency]
(iv) Since, ΔAQB ≅ ΔCPD
AQ = CP [CPCT]
(v) From (ii) and (iv), it is clear that APCQ has equal and opposite angles sides also equal opposite sides, Therefore, APCQ is a parallelogram.
10. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.21). Show that
(i) ΔAPB ≅ ΔCQD
(ii) AP = CQ
10. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.21). Show that
(i) ΔAPB ≅ ΔCQD
(ii) AP = CQ
Explanation:
(i) In ΔAPB and ΔCQD,
∠ABP = ∠CDQ (Alternate interior angles)
∠APB = ∠CQD (= 90o as AP and CQ are perpendiculars)
AB = CD (ABCD is a parallelogram)
ΔAPB ≅ ΔCQD [AAS congruency]
(ii) Since ΔAPB ≅ ΔCQD.
Therefore, AP = CQ [CPCT]
11. In ΔABC and ΔDEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F, respectively (see Fig. 8.22).
Show that
(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
(vi) ΔABC ≅ ΔDEF.
11. In ΔABC and ΔDEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F, respectively (see Fig. 8.22).
Show that
(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
(vi) ΔABC ≅ ΔDEF.
Explanation:
(i) AB = DE and AB || DE (Given)
Two opposite sides of a quadrilateral are parallel to each other and equal.
Therefore, quadrilateral ABED is a parallelogram
(ii) Again BC = EF and BC || EF.
Therefore, quadrilateral BEFC is a parallelogram.
(iii) Since ABED and BEFC are parallelograms.
⇒ AD = BE and BE = CF (Opposite sides of a parallelogram are equal)
AD = CF.
Also, AD || BE and BE || CF (Opposite sides of a parallelogram are parallel)
AD || CF
(iv) AD and CF are opposite sides of quadrilateral ACFD which are equal and parallel to each other. Therefore, it is a parallelogram.
(v) Since ACFD is a parallelogram
Therefore, AC || DF and AC = DF
(vi) In ΔABC and ΔDEF,
AB = DE (Given)
BC = EF (Given)
AC = DF (Opposite sides of a parallelogram)
ΔABC ≅ ΔDEF [SSS congruency]
12. ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.23). Show that
(i) ∠A = ∠B
(ii) ∠C = ∠D
(iii) ΔABC ≅ ΔBAD
(iv) diagonal AC = diagonal BD
[Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]
12. ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.23). Show that
(i) ∠A = ∠B
(ii) ∠C = ∠D
(iii) ΔABC ≅ ΔBAD
(iv) diagonal AC = diagonal BD
[Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]
Explanation:
To Construct: To draw a line through C parallel to DA intersecting AB produced at E.
(i) CE = AD (Opposite sides of a parallelogram)
AD = BC (Given)
BC = CE
⇒∠CBE = ∠CEB
Also,
∠A+∠CBE = 180° (Angles on the same side of transversal and ∠CBE = ∠CEB)
∠B +∠CBE = 180° ( As Linear pair)
⇒∠A = ∠B
(ii) ∠A+∠D = ∠B+∠C = 180° (Angles on the same side of transversal)
⇒∠A+∠D = ∠A+∠C (∠A = ∠B)
⇒∠D = ∠C
(iii) In ΔABC and ΔBAD,
AB = AB (Common)
∠DBA = ∠CBA
AD = BC (Given)
ΔABC ≅ ΔBAD [SAS congruency]
(iv) Diagonal AC = diagonal BD by CPCT because ΔABC ≅ ΔBAD.
Exercise 8.2
13.ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig 8.29). AC is a diagonal. Show that:
(i) SR || AC and SR = 1/2 AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.
Exercise 8.2
13.ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig 8.29). AC is a diagonal. Show that:
(i) SR || AC and SR = 1/2 AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.
Explanation:
(i) In ΔDAC,
S is the mid point of DA and R is the mid point of DC.
Therefore, by mid point theorem, SR || AC and SR = ½ AC
(ii) In ΔBAC,
Q is the mid point of BC and P is the mid point of AB
Therefore, by mid point theorem, PQ || AC and PQ = ½ AC
Also, SR = ½ AC
Hence, PQ = SR
(iii) SR || AC ———————- from (i)
and, PQ || AC ———————- from (ii)
⇒ SR || PQ – from (i) and (ii)
Also, PQ = SR
Hence, PQRS is a parallelogram.
14. 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.
14. 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.
Explanation:
Given ,
ABCD is a rhombus and P, Q, R, S are the mid-points of the sides AB, BC, CD and DA, respectively.
To Prove : PQRS is a rectangle.
Construction,
Join AC and BD.
Proof :
In ΔDRS and ΔBPQ,
DS = BQ (Halves of the opposite sides of the rhombus)
∠SDR = ∠QBP (Opposite angles of the rhombus)
DR = BP (Halves of the opposite sides of the rhombus)
ΔDRS ≅ ΔBPQ [SAS congruency]
RS = PQ [CPCT]———————- (i)
In ΔQCR and ΔSAP,
RC = PA (Halves of the opposite sides of the rhombus)
∠RCQ = ∠PAS (Opposite angles of the rhombus)
CQ = AS (Halves of the opposite sides of the rhombus)
ΔQCR ≅ ΔSAP [SAS congruency]
RQ = SP [CPCT]———————- (ii)
Now,
In ΔCDB,
Q and R are the mid points of BC and CD, respectively.
⇒ QR || BD
Also,
S and P are the mid points of AB and AD, respectively.
⇒ PS || BD
⇒ QR || PS
Therefore, PQRS is a parallelogram.
Also, ∠PQR = 90°
Now,
In PQRS,
RS = PQ and RQ = SP from (i) and (ii)
∠Q = 90°
Therefore, PQRS is a rectangle.
15. ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA, respectively. Show that the quadrilateral PQRS is a rhombus.
15. ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA, respectively. Show that the quadrilateral PQRS is a rhombus.
Explanation:
Given ,
ABCD is a rectangle and P, Q, R and S are mid-points of sides AB, BC, CD and DA, respectively.
Construction,
Join AC and BD.
To Prove : PQRS is a rhombus.
Proof:
In ΔABC
Q and P are the mid-points of BC and AB, respectively
PQ || AC and PQ = ½ AC (Midpoint theorem) — (i)
In ΔADC,
SR || AC and SR = ½ AC (Midpoint theorem) — (ii)
Hence, PQ || SR and PQ = SR
In quadrilateral PQRS one pair of opposite sides is equal and parallel to each other, hence it is a parallelogram.
, PS || QR and PS = QR (Opposite sides of parallelogram) — (iii)
Now,
In ΔBCD,
R and Q are mid points of side CD and BC, respectively.
QR || BD and QR = ½ BD (Midpoint theorem) — (iv)
AC = BD (Diagonals of a rectangle are equal) — (v)
From equations (i), (ii), (iii), (iv) and (v),
PQ = QR = SR = PS
Therefore, PQRS is a rhombus.
16. 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 (see Fig. 8.30). Show that F is the mid-point of BC.
16. 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 (see Fig. 8.30). Show that F is the mid-point of BC.
Explanation:
Given ,
ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD.
To prove : F is the mid-point of BC.
Proof :
BD intersected EF at G.
In ΔBAD,
E is the mid point of AD and also EG || AB.
Therefore, G is the mid point of BD (Converse of mid point theorem)
Now,
In ΔBDC,
G is the mid point of BD and also GF || AB || DC.
Therefore, F is the mid point of BC (Converse of mid point theorem).
17. In a parallelogram ABCD, E and F are the mid-points of sides AB and CD, respectively (see Fig. 8.31). Show that the line segments AF and EC trisect the diagonal BD.
17. In a parallelogram ABCD, E and F are the mid-points of sides AB and CD, respectively (see Fig. 8.31). Show that the line segments AF and EC trisect the diagonal BD.
Explanation:
Given ,
ABCD is a parallelogram. F and E are the mid-points of sides CD and AB , respectively.
To prove : AF and EC trisect the diagonal BD.
Proof :
ABCD is a parallelogram
AB || CD
Also, AE || FC
Now,
AB = CD (Opposite sides of parallelogram are equal)
⇒½ AB = ½ CD
⇒ AE = FC (E and F are midpoints of side AB and CD)
AECF is a parallelogram (AE and CF are equal and parallel to each other)
AF || EC (Opposite sides of a parallelogram)
Now,
In ΔDQC,
F is mid point of side DC and FP || CQ (as AF || EC).
P is the mid-point of DQ (Converse of mid-point theorem)
⇒ DP = PQ — (i)
Similarly,
In ΔAPB,
E is midpoint of side AB and EQ || AP (as AF || EC).
Q is the mid-point of PB (Converse of mid-point theorem)
⇒ PQ = QB — (ii)
From (i) and (i),
DP = PQ = BQ
Therefore, the line segments AF and EC trisect the diagonal BD.
18. Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
18. Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
Explanation:
Let ABCD be a quadrilateral and P, Q, R and S be the mid points of AB, BC, CD and DA, respectively.
Now,
In ΔACD,
S and R are the mid points of DA and CD, respectively.
Therefore, SR || AC.
Similarly ,
PQ || AC,
PS || BD and
QR || BD
Therefore PQRS is parallelogram.
Hence, PR and QS are the diagonals of the parallelogram PQRS. Therefore, they will bisect each other.
19. 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 = ½ AB
19. 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 = ½ AB
Explanation:
(i) In ΔACB,
M is the midpoint of AB and MD || BC
D is the midpoint of AC (Converse of mid point theorem)
(ii) ∠ACB = ∠ADM (Corresponding angles)
Also, ∠ACB = 90°
∠ADM = 90° and MD ⊥ AC
(iii) In ΔAMD and ΔCMD,
AD = CD (D is the midpoint of side AC)
∠ADM = ∠CDM (Each 90°)
DM = DM (common)
ΔAMD ≅ ΔCMD [SAS congruency]
AM = CM [CPCT]
Also, AM = ½ AB (M is midpoint of AB)
Therefore, CM = MA = ½ AB
Also Read: Quadrilateral Class 9 Extra Questions
