KC Sinha: Exercise 9.2 - Mathematics Solution Class 9 Chapter 9 Triangles

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EXERCISE 9.2

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Type 1

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QUESTION 1

Fill up the blanks so that the statements given below are true :
(i) In a triangle, ,difference of any two sides is __ than the third side
(ii) In a triangle, sum of any two sides is ___ than the third side
(iii) In a triangle, side opposite to larger angle is ___
(iv) In a triangle, side opposite to the smallest angle is __
(v) In a triangle, angle Opposite to the greatest side is __
(vi) Of all the line segments that can be drawn to a given line from a point not lying on it, the smallest is ___
(vii) In a right angled triangle, hypotenuse is the ___ side
(viii) Perimeter of a triangle is ___  than sum of its three medians.
(ix) The sum of three altitudes of a triangle is __ than its perimeter
(x) In a ΔABC , if ∠B=120° , then side opposite to ∠B will be __
Sol :

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QUESTION 2

Which of the following statements are true (T) and which are false (F) :
(i) Of all the line segments that can be drawn to a given line from a point not lying on it, perpendicular line segment is the shortest one.

(ii) In a triangle, sum of two sides is smaller than the third side.
(iii) In a triangle,sum of two angles is greater than the third angle.
(iv) In an acute angled triangle, sum of two angles is greater than the third angle.
(v) In a triangle, difference of two sides is equal to the third side.

(vi) In a triangle. sum of two sides is greater than the third side.
Sol :

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QUESTION 3

See f‌igure below and fill up the blanks with the signs of inequality and equality.

(i) ∠BDA __ ∠DCB
(ii) ∠BDA __ ∠DBC
(iii) If ∠ABC>∠ACB, then AC__AB
(iv) If ∠ABC=∠ACB, then AC__AB
(v) If ∠ABD=∠ADB, then AB__AD
Sol :

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QUESTION 4

(i) In a triangle, two sides are 5cm and 3cm , then can third side be 1.5 cm ?
(ii) If AB=4 cm, BC=3 cm and CA=8 cm, then is construction of ΔABC possible ?

(iii) In a ΔABC , if ∠A=60° , ∠B=75° and ∠C=45° , then write,
(a) Greatest side (b) Smallest side
(iv) Of all the line segments that can be drawn to a line from a point outside the line which is the smallest line segment ?
(v) How many perpendiculars can be drawn on a line from a point outside the line ?
Sol :

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QUESTION 5

In an examination. there was a question: construct a triangle whose three sides are of length 3.6 cm, 4.6 cm and 8.4 cm respectively. Was the question correct ? Explain giving reasons.
Sol :

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QUESTION 6

Can a ΔABC be constructed, if AB=6cm , BC=4cm and CA=3.2cm ? Given reasons.
Sol :

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TYPE 2

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QUESTION 7

What can be the values in integer, of the third side of a triangle, if its two sides are as follows ?
(a) 2 and 6
(b) 7 and 7
(c) 4 and 8
(d) 3 and 9
Sol :

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TYPE 3

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QUESTION 8

In figure below, sides PQ and PR are produced and ∠SQR<∠TRQ . Show that PR>PQ

Sol :

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QUESTION 9

In figure below, AB>AC and D is a point on BC. Show that AB>AD

Sol :

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QUESTION 10

In figure below, PR>PQ and PS is a bisector of ∠P . Show that x>y

Sol :

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QUESTION 11

In triangle (Fig. below) , ∠B<∠A and ∠C<∠D , then prove that AD<BC.

Sol :

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QUESTION 12

Prove that the sum of any two sides of a triangle is greater than twice the median of the third side.

<fig to be added>
Sol :

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QUESTION 13

Prove that , sum of three sides of a triangle is greater than the sum of its medians.
Sol :

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QUESTION 14

In the adjoining figure, from point P not lying on a line m , line segments are drawn to m , PD being the shortest one. If B and C be the points on m such that D is the mid point of BC . Prove that PB=PC
[Hint: Let m be a line and P be a point not lying on m. On the line m , B and C are two points such that BD=DC.
To prove : PB=PC
Proof: Of line segments drawn from P to m. PD is the smallest,
∴PD⊥m
That is , ∠PDB=∠PDC=90°
Now, prove congruence of ΔPBD and ΔPCD ]

Sol :

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QUESTION 15

In a ΔABC, internal bisectors of ∠B and ∠C meet at point O. If AC>AB, then prove that OC>OB.
[Hint: AC>AB
⇒∠ABC>∠ACB
⇒1/2∠ABC>1/2∠ACB
⇒∠OBC>∠OCB
⇒∠OC<∠OB]

Sol :

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QUESTION 16

Prove that the sum of distances of vertices of a triangle from any point inside the triangle is greater than its half perimeter.
Sol :

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QUESTION 17

In adjoining f‌igure, AP⊥l and and PR>PQ, prove that AR>AQ
[Hint: Take PQ=PS and join AS
∠1=∠3>∠2
∠1>∠2
AR>AQ]

Sol :

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QUESTION 18

S is any point in the interior of ΔPQR, show that 
SQ+SR<PQ+PR
[Hint: Join QS and produce this to cut PR at T]

Sol :

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QUESTION 19

In ΔPQR, S is a point on the side QR. Show that PQ+QR+RP>2PS

Sol :

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QUESTION 20

O is a point inside a quadrilateral ABCD which is not at the point of intersection of diagonals . Prove that
OA+OB+OC+OD>AC+BC
Sol :

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QUESTION 21

In figure , AB=AC , then prove that AF>AE
[Hint: AB=AC
⇒From ΔEBD, ∠4>∠2
⇒∠4>∠3 [AB=AC, ∠2=∠3]
⇒∠4>∠5
⇒In ΔAEF; AF>AE
]
<fig to be added>
Sol :

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QUESTION 22

In figure below, T is a point on the side QR of ΔPQR, S is a point such that RT=ST. Prove that PQ+PR>QS

Sol :

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QUESTION 23

In the f‌igure below, AC>AB and D is a point on AC such that AB=AD. Prove that CD<BC
<fig to be added>
[Hint: AB=AD
In ΔABC, AB+BC>AC
AB+BC>AD+CD
BC>CD
CD<BC [AB=AD]
]
Sol :