Class 9 Maths Important Questions for Quadrilaterals
Given below are the Class 9 Maths Extra Questions for Quadrilaterals
a. True and False
b. Hard problems
c. Multiple choice questions
d. Long answer questions
e. Fill in the blanks
True and False
Question 1 Which are these is true or false about parallelogram
The diagonals of a parallelogram bisect each other
In a parallelogram, opposite sides and angle are equal.
A diagonal of a parallelogram divides it into two congruent triangles
The bisectors of the angles of parallelogram create a rectangle
Sum of all the internal angles is 3600
Sum of all the exterior angles is 1800
Square, rectangle and rhombus are all parallelogram
True. This can be proved easily using SSS congruence
True
True. This can easily proved by drawing one diagonal and summing all the angles based on triangle angle sum
False
True
True
Question 2 True or False statement
All the angles of the quadrilateral are obtuse
Diagonal of the rhombus are equal and perpendicular to each other
Diagonal of the square are equal and bisect each other at right angle
Out of four points A,B,C,D in place, there are collinear. A quadrilateral can be formed from these points
Trapezium, in which the sides that are not parallel are equal in length and angles formed by parallel sides are equal, such trapezium is called isosceles trapezium
Question 3
ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD
Which of the following is true based on given information
a. $AP=CQ$
b. $QD=PB$
c. $DP=QB$
d. $\Delta PAD \cong \Delta QCB$ Solution
All are correct
In Triangle $ \Delta PAD \; and \; \Delta QCB$
AD=CB
$\angle P= \angle Q=90^0$
$\angle CBQ=\angle ADP$ ( Alternate interior angles of AB||CD)
So AAS congruence
Also as they are congruent, we get AP=CQ and DP=QB
Now Lets see the triangles $\Delta APB \; and \; \Delta CQD$
AB=CD
$\angle P= \angle Q=90^0$ (Alternate interior angles of AB||CD)
So QD=PB
Question 4
The angles of the quadrilateral are in the ratio 2:5:4:1?
Which of the following is true?
a. Largest angle in the quadrilateral is $150^0$
b. Smallest angle is $30^0$
c.The second largest angle in the quadrilateral is $80^0$
d.None of these Solution
(a) and (b)
Angles are $2x, 5x,4x,x$
Now
$2x+5x+4x+x=360$
Or $x=30$
Angles are $30^0, 60^0,120^0,150^0$
Question 5
Two adjacent angles in a parallelogram are in the ratio 2:4. Find the values?
a. 80,100
b. 40,140
c. 60,120
d. None of the above Solution
(c)
Adjacent angles
$2x+4x=180$
$x=30$
60,120 are adjacent angles
Question 6
ABCD is a trapezium with AB =10cm, AD=5 cm, BC=4 cm and DC =7 cm?
Find the area of the ABCD
a. 34 cm2
b. 28cm2
c. 20 cm2
d. None of these Solution
(a)
BC is the altitude between the two parallel sides AB and DC
So Area of trapezium will be given by
$A=\frac {1}{2}BC(AB+DC)=34cm^2$
Question 7
PQRS is a quadrilateral whose diagonal bisect each other at right angles
a. PQRS is a Square
b. PQRS is a rectangle
c. PQRS is a rhombus
d. None of these Solution
(c)
Question 8
ABCD is a trapezium where AB||DC. BD is the diagonal and E is the mid point of AD. A line is draw from point E parallel to AB intersecting BC at F. Which of these is true?
a. BF=FC
b. EA=FB
c. CF=DE
d. None of these Solution
(a)
Let’s call the point of intersection at diagonal as G
Then in triangle DAB
EG||AB and E is the mid point of DA,So by converse of mid point theorem,
G is the mid point of BD
Now in triangle DBC
GF||CD
G is the mid point of DB
So by converse of mid point theorem
F is the mid point of BC
Question 9
ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively
a. PQRS is a rectangle
b. PQRS is a parallelogram
c. PQRS is a rhombus
d. None of these Solution
(c)
Question 10
In a parallelogram PQRS, The bisector of angles P and Q meet at point O as shown in below figure. What is the angle O?
a. 80
b. 90
c. 45
d. None of the above Solution
(b)
Match the column
Question 11
Rhombus
Is a quadrilateral with only one pair of parallel sides.
Rectangles
Is a quadrilateral whose Two pairs of adjacent sides of a kite are equal, and one pair of opposite angles is equal. Diagonals intersect at right angles. One diagonal is bisected by the other.
Kite
Is a quadrilateral whose all the sides are equal and opposite sides are parallel. Opposite angles are equal.
Right-angled trapezoid
Is a quadrilateral whose opposite sides of a rectangle are parallel and equal. All angles are 90º.
Isosceles trapezoid
A trapezoid having two right angles
Trapezoid
Is a trapezoid whose non parallel sides are equal
Long answer type
Question 12.
Show that the quadrilateral formed by joining the mid- points of adjacent sides of rectangle is a rhombus. Solution
The figure is shown as below
To Prove: quadrilateral PQRS is a rhombus
Proof:
In Δ ABC
P and Q are mid points of sides AB and BC
By Mid point theorem
PQ= 1/2 AC and PQ || AC -(X)
In Δ ACD
S and R are mid points of sides AD and DC
By Mid point theorem
SR= 1/2 AC and SR || AC --(Y)
From (X) and (Y), we have
PQ=SR
PQ ||SR
Hence PQSR is a parallelogram
Now in Δ BCD
Q and R are mid points of sides BC and DC
By Mid point theorem
QR= 1/2 BD
Nowe AC = BC
Hence
PQ=SR=QR
Now a parallelogram whose adjacent sides are equal is a rhombus.
Hence proved
Question 13
P, Q, R and S are respectively the mid- point of sides AB, BC, CD and DA of a quadrilateral ABCD such that AC=BD. Prove that PQRS is a rhombus. Solution
Given AC=BD
Proof:
In Δ ABC
P and Q are mid points of sides AB and BC
By Mid point theorem
PQ= 1/2 AC and PQ || AC -(X)
In Δ ACD
S and R are mid points of sides AD and DC
By Mid point theorem
SR= 1/2 AC and SR || AC --(Y)
From (X) and (Y), we have
PQ=SR= 1/2 AC ----(1)
Similarly in Δ BCD
Q and R are mid point of BC and CD
By Mid point theorem
QR= 1/2 BD
Similarly in Δ ADB
S and P are mid point of AD and AB
By Mid point theorem
SP= 1/2 BD
Therefore
SP=QR=1/2 BD ---(2)
AC=BD
So from (1) and (2)
PQ=SR=SP=QR
Hence PQRS is a rhombus
Question 14.
Prove that the quadrilateral formed (if possible) by the internal angular bisectors of any quadrilateral is cyclic. Question 15.
PQ and RS are to equal and parallel line segments. Any point M not lying on PQ or RS is joined to Q and S and lines through P parallel to QM and through R parallel to SM meet at N. Prove that the line segments MN and PQ are equal and parallel to each other. Question 16.
l, m and n are three parallel lines intersected by transversal's p and q such that l, m and n cut off equal intercepts AB and BC on p. Show that l, m and n cut off equal intercepts DE and EF on q also. Solution
Given
AB=BC
To Prove:
DE=EF
Proof:
Let us join A to F intersecting m at G
The trapezium ACFD is divided into two triangles namely Δ ACF and &Delta AFD
In Δ ACF, it is given that B is the mid-point of AC (AB = BC)
and BG || CF (since m || n).
So, G is the mid-point of AF (by Mid Point Theorem)
Now, in Δ AFD, we can apply the same argument as G is the mid-point of AF,
GE || AD and so by by Mid Point Theorem, E is the mid-point of DF,
i.e., DE = EF.
In other words, l, m and n cut off equal intercepts on q also.
Question 17.
In a quadrilateral ABCD, CO and DO are bisectors of $\angle C$ and $\angle D$ respectively. Prove that $\angle COD = \frac {1}{2} (\angle A + \angle B)$ Question 18.
ABC is a triangle. D is a point on AB such that AD = ¼ AB and E is the point on A such that AE = ¼ AC. Prove that DE = ¼ BC. Question 19.
If the diagonals of a parallelogram are equal, then show that it is a rectangle. Question 20.
In a parallelogram, show that the angle bisectors of two adjacent angles intersect at right angles. Question 21.
Show that the line segment joining the mid- point of any two sides of a triangle is parallel to third side and is equal to half of it. Question 22.
A diagonal of a rectangle is inclined to one side of a rectangle at $25^0$. Find the acute angles between the rectangle diagonals. Question 23.
ABCD is a parallelogram, AD is produced to E so that DE = DC and EC produced meets AB produced in F. Prove that BF = BC. Question 24.
ABCD is a parallelogram P is a point on AD such that AP = 1/3 AD and Q is a point on BC such that CQ = 1/3 BC. Prove that AQCP is a parallelogram. Question 25.
P is the mid- point of side AB of a parallelogram ABCD. A line through B parallel to PD meets DC at Q and AD produced at R. Prove that
AR = 2BC
BR = 2BQ.
Question 26
ABCD is a kite having AB = AD and BC = CD. Prove that the figure formed by joining the mid- points of the side, in order, is a rectangle.
Summary
This Quadrilaterals class 9 extra questions with answers is prepared keeping in mind the latest syllabus of CBSE . This has been designed in a way to improve the academic performance of the students. If you find mistakes , please do provide the feedback on the mail.
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a. BF=FC
a. 80
