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CBSE Questions for Class 9 Maths Triangles Quiz 5 - MCQExams.com
CBSE
Class 9 Maths
Triangles
Quiz 5
The distance from town A to town B is five miles. C is six miles from B. Which of the following could be the distance from A to C?
I.
11
II.
1
III.
7
Report Question
0%
I only
0%
II only
0%
I and II only
0%
II and III only
0%
I, II or III
Explanation
The distance from two A nad B is five miles. C is six miles from B To find distance between A to C there are three possibilities.
Case:
1
In triangle
A
B
C
,
A
B
=
5
and
B
C
=
6
Here , the length of AC must be less than the sum of other two sides and greater than the difference of the other sides
6.5
<
6
+
5
⇒
1
<
A
c
<
11
Thus
A
C
=
7
is a possible
∴
distance from
A
to
C
could be
11
,
1
or
7
miles.
Two sides of a triangle have lengths
7
and
9
. Which of the following could not be the length of the third side?
Report Question
0%
4
0%
5
0%
7
0%
11
0%
16
Explanation
An important rule to remember about triangles is called the third side rule:
The length of the third side of a triangle is less than the sum of the lengths of the other two sides and greater than the (positive) difference of the lengths of the other two sides.
For this triangle, the length of the third side must be greater than
9
−
7
=
2
97=2
and less than
9
+
7
=
16
9+7=16
. All the answers are possible except for answer E, which is equal to
16
16
but not less than
16
16
.
Which of the following sets of measurements can be used to construct a triangle?
Report Question
0%
4
cm
,
5
cm
,
6
cm
0%
4
cm
,
3
cm
,
8
cm
0%
5
cm
,
6
cm
,
12
cm
0%
6
cm
,
3
cm
,
10
cm
Explanation
Because the sum of the length of any
2
sides of the triangle should be greater than the third side, which is only satisfied by option
A
.
(
4
+
5
)
>
6
(
4
+
6
)
>
5
(
5
+
6
)
>
4
In the figure above,
A
B
C
D
is a rectangle. If
A
D
=
6
, which of the following could be the length of
¯
A
C
?
Report Question
0%
2
0%
4
0%
5
0%
6
0%
7
Explanation
In triangle
A
C
D
,
angle
A
D
C
is greater than the other two angles
By sine rule , we get
A
C
A
D
=
∠
(
A
D
C
)
∠
(
D
C
A
)
We know that
∠
(
A
D
C
)
>
∠
(
D
C
A
)
Therefore
A
C
>
A
D
=
6
Therefore option
E
is correct
In a triangle with sides of
7
and
9
, the third side must be
Report Question
0%
more than
16
0%
between
7
and
9
0%
between
2
and
16
0%
between
7
and
16
0%
between
9
and
16
Explanation
Given that two sides of triangle are
7
,
9
Let the third side be
x
We have
7
+
9
>
x
and
x
+
9
>
7
and
x
+
7
>
9
and
x
>
0
We get
x
<
16
and
x
>
2
Therefore the third side will lie between
2
and
16
.
Mark the triplet that can be the lengths of the sides of a triangle.
Report Question
0%
2
,
3
,
5
0%
1
,
4
,
2
0%
7
,
4
,
4
0%
5
,
6
,
12
0%
9
,
20
,
8
Explanation
for triplet to be the length of triangle, it must satisfy triangle property.
sum of any two sides must be greater than third side
So,
7
,
4
,
4
is correct answer.
If we consider remaining options, observe that sum of two sides is less than third side, so they cannot form triangle.
△
P
Q
R
is right angled at
Q
,
P
R
=
5
c
m
and
Q
R
=
4
c
m
. If the lengths of sides of another triangle
A
B
C
are
3
c
m
,
4
c
m
,
5
c
m
, then which one of the following is correct?
Report Question
0%
Area of
△
P
Q
R
is double that of
△
A
B
C
.
0%
Area of
△
A
B
C
is double that of
△
P
Q
R
.
0%
√
B
=
√
Q
2
0%
Both triangles are congruent.
Explanation
Given:
Δ
P
Q
R
is right angled at
Q
,
P
R
=
5
c
m
,
Q
R
=
4
c
m
.
Length of sides of another triangle are
3
c
m
,
5
c
m
,
4
c
m
.
According to question,
Ref. image
Applying Pythagoras theorem in
Δ
P
Q
R
,
H
2
=
P
2
+
B
2
5
2
=
P
2
+
4
2
25
−
16
=
P
2
9
=
P
2
P
=
3
c
m
,
P
Q
=
3
c
m
Now we can see that both the triangles have equal sides.
i.e.
P
Q
=
A
B
=
3
c
m
P
R
=
A
C
=
5
c
m
Q
R
=
B
C
=
4
c
m
It means by SSS congruence property both triangles are congruent.
{SSS congruence property: Two triangles are congruent if the three sides of one triangle are respectively equal to the three sides of other triangle.}
i.e.
Δ
P
Q
R
≅
Δ
A
B
C
The correct option is (D). i.e. both triangles are congruent.
In quadrilateral
A
C
B
D
,
A
C
=
A
D
and
A
B
bisects
∠
C
A
D
.
Show that
△
A
B
C
≅
△
A
B
D
. Can you say that
B
C
=
B
D
?
Report Question
0%
True
0%
False
Explanation
In
△
A
B
C
and
△
A
B
D
,
∠
C
A
B
=
∠
B
A
D
[Given]
A
C
=
A
D
[Given]
A
B
=
A
B
[Common]
So, by
S
A
S
rule of congruence,
△
A
B
C
≅
△
A
B
D
As corresponding sides in congruent triangles are equal so,
B
C
=
B
D
.
In
△
A
B
C
, the bisector of
∠
A
intersects
¯
B
C
at a point
D
. Then:
Report Question
0%
B
D
×
A
C
=
B
C
×
A
B
0%
B
D
×
A
B
=
D
C
×
A
C
0%
A
C
×
A
B
=
D
C
×
B
C
0%
B
D
×
A
C
=
D
C
×
A
B
Explanation
In
△
A
B
C
, the bisector of
∠
A
intersects
¯
B
C
at
D
.
∴
A
D
bisects
B
C
∴
B
D
=
D
C
........
(
i
)
In triangles,
A
B
D
and
A
D
C
B
D
=
C
D
......... From
(
i
)
∠
B
A
D
=
∠
D
A
C
....... (Given)
A
D
=
A
D
........... (Common side)
∴
△
A
B
D
≅
△
A
D
C
........ [By S.A.S criterion]
⟹
A
B
=
A
C
Thus,
B
D
×
A
C
=
D
C
×
A
B
Which of the following set of measurements will form a triangle
Report Question
0%
11
c
m
,
4
c
m
,
6
c
m
0%
13
c
m
,
14
c
m
,
25
c
m
0%
8
c
m
,
4
c
m
,
3
c
m
0%
5
c
m
.16
c
m
.5
c
m
Explanation
Triangle Inequality Theorem states that the sum of two side lengths of a triangle is always greater than the third side.
If this is true for all three combinations of added side lengths.
i.e.
a
+
b
>
c
,
b
+
c
>
a
a
nd
c
+
a
>
b
then the lengths form a triangle
(A)
11
+
4
=
15
>
6
And
4
+
6
=
10
≯
11
So, it does not form a triangle
(B)
13
+
14
=
27
>
25
And
14
+
25
=
39
>
13
And
25
+
13
=
38
>
14
So, it forms a triangle
(C)
8
+
4
=
12
>
3
And
8
+
3
=
11
>
4
And
4
+
3
=
7
≯
8
So, it does not form a triangle
(D)
5
+
16
=
21
>
5
And
5
+
5
=
10
≯
16
So, it does not form a triangle.
Hence option B is the correct answer
The two triangles in the figure are congruent by the congruence theorem. Here, it is given
O
Q
=
O
R
. Which of the following condition, along with the given condition, is sufficient to prove that the two triangles are congruent to each other?
Report Question
0%
∠
P
=
∠
S
0%
∠
Q
=
∠
R
0%
O
P
=
O
S
0%
P
Q
=
S
R
Explanation
Given
O
Q
=
O
R
and
△
P
O
Q
≅
△
R
O
S
We know that, congruent parts of congruent triangles are congruent
∠
P
O
Q
≅
∠
R
O
S
(vertically opposite angles)
If
O
P
=
O
S
then by using the (SAS) congruent, we can conclude the congruency of two triangles.
Hence, option C is sufficient to prove the congruency.
△
F
G
C
is an isosceles triangle, which of the following method will prove
△
A
B
C
congruent to
△
D
E
F
?
Report Question
0%
SSS
0%
RHS
0%
AAS
0%
SAS
Explanation
In
△
A
B
C
and
△
D
E
F
Given,
A
B
=
D
E
∠
A
B
C
=
∠
D
E
F
=
90
∘
△
F
C
G
is isoceles,
G
F
=
G
C
⟹
F
D
=
C
A
...... (as
A
B
|
|
D
E
,
G
D
should be equal to
G
A
)
A
C
=
F
D
Hence,
△
A
B
C
≅
△
D
E
F
by
R
H
S
p
o
s
t
u
l
a
t
e
Triangles with sides
3
cm,
4
cm and
5
cm is possible.
Report Question
0%
True
0%
False
Explanation
Yes, as
3
2
+
4
2
=
5
2
Consider isosceles triangle
A
B
C
, in which
∠
A
B
C
=
∠
A
C
B
,then which of the two sides are similar?
Report Question
0%
A
B
=
A
C
0%
A
B
=
B
C
0%
A
C
=
B
C
0%
All of the above
Explanation
Here,
△
A
B
C
is an isosceles triangle.
Given,
∠
A
B
C
=
∠
A
C
B
.
We know, by isosceles triangle property, angles opposite to equal sides are equal.
Thus, if
∠
A
B
C
=
∠
A
C
B
, then
A
B
=
A
C
(as
∠
A
C
B
is opposite to side
A
B
and
∠
A
B
C
is opposite to side
A
C
).
Hence, option
A
is correct.
To prove that
Δ
D
E
F
and
Δ
A
B
C
are congruent by SAS, what additional information is needed?
Report Question
0%
E
F
≅
B
C
0%
∠
D
F
E
≅
∠
A
B
C
0%
D
E
≅
A
B
0%
∠
D
F
E
≅
∠
A
C
B
Explanation
Given
F
D
=
C
A
and
∠
D
=
∠
A
Now for triangles to be congruent by
S
A
S
we need a third side that include the given angles with the given equal sides.
So
D
E
=
A
B
⟹
D
E
≅
A
B
Hence, option
C
is correct.
Consider isosceles triangle
A
B
C
, in which
∠
A
B
C
=
∠
A
C
B
,
A
B
=
2
B
C
and
A
B
=
8
c
m
. What is the perimeter of the
△
A
B
C
?
Report Question
0%
24
c
m
0%
32
c
m
0%
20
c
m
0%
18
c
m
Explanation
△
A
B
C
is an isosceles triangle.
Gi en that
∠
A
B
C
=
∠
A
C
B
,
A
B
=
2
B
C
and
A
B
=
8
cm$$
So,
A
B
=
A
C
=
8
cm
⇒
B
C
=
A
B
2
=
4
cm
Perimeter of
△
A
B
C
=
8
+
8
+
4
=
20
cm.
Here,
△
A
B
C
is an isosceles triangle.
Given,
∠
A
B
C
=
∠
A
C
B
,
A
B
=
2
B
C
and
A
B
=
8
c
m
.
We know, by isosceles triangle property, angles opposite to equal sides are equal.
Then,
So,
A
B
=
A
C
=
8
c
m
.
Since
A
B
=
2
B
C
,
⇒
B
C
=
A
B
2
=
4
c
m
.
Thus, p
erimeter of
△
A
B
C
=
8
+
8
+
4
=
20
c
m
.
Hence, option
C
is correct.
State Whether the following statement is True or False:
In the figure,
C
is the mid-point of
A
B
,
∠
B
A
D
=
∠
C
B
E
,
∠
E
C
A
=
∠
D
C
B
then
D
A
=
E
B
.
Report Question
0%
True
0%
False
Explanation
Given
∠
P
=
∠
Q
⟹
∠
E
C
A
=
∠
D
C
B
.
Now adding
∠
X
both side we get:
∠
P
+
∠
X
=
∠
Q
+
∠
X
........
[
1
]
.
Now in triangles
Δ
A
C
D
and
Δ
B
C
E
,
∠
A
=
∠
B
........[given]
⟹
∠
B
A
D
=
∠
A
B
E
A
C
=
B
C
...........[
C
is midpoint of
A
B
]
∠
A
C
D
=
∠
B
C
E
.......
[
1
]
So,
Δ
A
C
D
≅
Δ
B
C
E
......[by
A
S
A
congruency criterion]
Hence by
C
P
C
T
, the corresponding parts are also equal.
Then,
A
D
=
B
E
.
Therefore, the given statement is true.
Hence, option
A
is correct.
In the figure,
∠
B
C
D
=
∠
A
D
C
and
∠
A
C
B
=
∠
B
D
A
.
Report Question
0%
∠
B
=
∠
B
0%
∠
A
=
∠
D
0%
∠
A
=
∠
B
0%
∠
C
=
∠
D
Explanation
In
△
A
C
D
and
△
B
D
C
∠
B
C
D
=
∠
A
D
C
[Given].
Given,
∠
A
C
B
=
∠
B
D
A
.
Adding these two above equation , we get,
∠
B
C
D
+
∠
A
C
B
=
∠
A
D
C
+
∠
B
D
A
⟹
∠
A
C
D
=
∠
B
D
C
.
Also,
C
D
=
C
D
[Common]
△
A
C
D
≅△
B
D
C
.....(By
A
S
A
congruency criterion).
Therefore, by
C
P
C
T
rule, the corresponding parts are equal.
Then,
A
D
=
B
C
and
∠
A
=
∠
B
.
Hence, option
C
is correct.
In a quadrilateral
A
C
B
D
,
A
C
=
A
D
and
A
B
bisect
∠
A
, then:
Report Question
0%
B
C
≅
A
D
0%
B
D
≅
A
C
0%
B
C
≅
B
D
0%
A
B
≅
A
C
Explanation
In
△
A
B
C
and
△
A
B
D
,
A
B
=
A
B
(Common)
∠
C
A
B
=
∠
D
A
B
(
A
B
bisect
∠
A
)
A
C
=
A
D
(Given)
By using
S
A
S
rule of congruence,
△
A
B
C
≅
△
A
B
D
⇒
B
C
≅
B
D
[by
C
P
C
T
]
Therefore, option
C
is correct.
State whether the following statement is True or False?
Two sides
A
B
,
B
C
and median
A
M
of one triangle
A
B
C
are respectively equal to sides
P
Q
and
Q
R
and median
P
N
of
△
P
Q
R
. then
△
A
B
M
≅
△
P
Q
N
Report Question
0%
True
0%
False
Explanation
In
△
A
B
M
and
△
P
Q
N
A
B
=
P
Q
B
C
=
Q
R
A
M
=
P
N
A
M
is the median of
△
A
B
C
So,
B
M
=
C
M
=
1
2
B
C
P
N
is the median of
△
P
Q
R
So,
Q
N
=
R
N
=
1
2
Q
R
Therefore,
B
M
=
Q
N
Hence,
△
A
B
M
≅△
P
Q
N
{By SSS congruence}
In given figure
△
P
Q
R
≅
△
X
Y
Z
by _______ congruency rule.
Report Question
0%
S
S
S
0%
A
A
A
0%
S
A
S
0%
A
S
A
Explanation
In
△
P
Q
R
and
△
X
Y
Z
,
∠
Q
P
R
=
∠
Y
X
Z
(given)
P
Q
=
X
Y
(given)
∠
P
Q
R
=
∠
X
Y
Z
(given)
Therefore,
△
P
Q
R
≅
△
X
Y
Z
by
A
S
A
rule of congruency.
State the following statement is True or False:
In an isosceles triangle any two angles of a triangle are same.
Report Question
0%
True
0%
False
Explanation
We know, by isosceles triangle property, the angles opposite to equal sides are also equal.
Then,
in an isosceles triangle only the angles opposite to equal side are equal.
Hence, in an isosceles triangle, any two angles of the triangle are not same.
Therefore, the statement is false and option
B
is correct.
In the given figure,
A
L
∥
D
C
,
E
is mid point of
B
C
, then
△
E
B
L
≅
△
E
C
D
.
Report Question
0%
True
0%
False
Explanation
In
△
E
B
L
and
△
E
C
D
, we have,
∠
B
E
L
=
∠
C
E
D
....{vertical oppsoite angle}
B
E
=
C
E
.....{
E
is the mid-point of
B
C
}
∠
E
B
L
=
∠
E
C
D
....{alternate interior angle}.
Therefore,
△
E
B
L
≅△
E
C
D
...{By
A
S
A
Congruency criterion}.
Hence, the given statement is true.
Therefore, option
A
is correct.
Which congruence criteria can be used to state that
△
XOY
≅
△
POQ?
Report Question
0%
ASA
0%
SAS
0%
SSS
0%
RHS
Explanation
In
△
X
O
Y
and
△
P
O
Q
⇒
∠
Y
X
O
=
∠
Q
P
O
=
65
o
[Given]
⇒
O
X
=
O
P
[Given]
⇒
∠
X
O
Y
=
∠
P
O
Q
[Vertically opposite angles]
∴
△
X
O
Y
≅
△
P
O
Q
[By ASA criteria]
In Fig if
∠
C
>
∠
A
,
∠
D
>
∠
E
then
Report Question
0%
A
E
>
C
D
0%
A
E
<
C
D
0%
A
B
<
B
C
0%
B
E
<
B
D
Explanation
Given,
∠
C
>
∠
A
⟹
A
B
>
B
C
------ Side opposite to largest angle is longest
Similarly,
∠
D
>
∠
E
⟹
E
B
>
B
D
From above,
(
A
B
+
E
B
)
>
(
B
C
+
B
D
)
⟹
A
E
>
C
D
Option A
In the given figure, which of the following is correct?
Report Question
0%
Δ
P
Q
R
≅
Δ
R
S
P
0%
Δ
P
Q
R
≅
Δ
S
P
R
0%
Δ
P
Q
R
≅
Δ
R
P
S
0%
Δ
P
Q
R
≅
Δ
P
S
R
Explanation
In
△
P
Q
R
and
△
R
S
P
∠
Q
P
R
=
∠
S
R
P
=
45
o
P
Q
=
R
S
=
5.5
c
m
P
R
=
R
P
(common)
△
P
Q
R
≅
△
R
S
P
[By SAS congruency]
Hence option (A) is correct
In given figure,
CF and AE are equal perpendiculars on BD, BF = FE = ED.
∠
BAE = ______.
Report Question
0%
∠
BCD
0%
∠
CBA
0%
∠
ADC
0%
∠
DCF
Explanation
In
△
A
B
E
and
△
C
D
F
⇒
A
E
=
C
F
[given]
⇒
∠
A
E
B
=
∠
C
F
D
[given]
⇒
B
F
+
F
E
=
D
E
+
E
F
⇒
B
E
=
D
F
∴
△
A
B
E
≅
△
C
D
F
[By SAS criteria]
∴
∠
B
A
E
=
∠
D
C
F
[By CPCT]
In the given figure, the line segment
A
B
is parallel to another line segment
R
S
and
O
is the midpoint of
A
S
,
Then
Δ
A
O
B
≅
Δ
S
O
R
?
Report Question
0%
True
0%
False
Explanation
A
O
=
O
S
∠
R
O
S
=
∠
A
O
B
(
V
e
r
t
i
c
a
l
l
y
O
p
p
o
s
i
t
e
A
n
g
l
e
)
∠
R
S
A
=
∠
B
A
S
(
A
l
t
e
r
n
a
t
i
n
g
A
n
g
l
e
)
Hence, According to ASA Congruence Criterion
△
A
O
B
≅
△
S
O
R
In the figure given below,
B
A
∥
D
F
and
C
A
∥
E
G
and
B
D
=
E
C
, then
B
G
=
D
F
.
Report Question
0%
True
0%
False
In the figure ,
∠
A
=
∠
C
and
A
B
=
B
C
. Then
Δ
A
B
D
≅
Δ
C
B
E
Report Question
0%
True
0%
False
Explanation
Since,
∠
B
is common to both Triangles
Hence,
∠
B
=
∠
B
∠
A
=
∠
C
A
B
=
A
C
From ASA Congruency Criterion
△
A
B
D
≅
△
C
B
E
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