NCERT Solutions Class 9 Mathematics Chapter 7 Triangles Download In Pdf

Chapter 7 Triangles Download in Pdf

EXERCISE 7.1

**Question 1.** In quadrilateral ACBD,
AC = AD and AB bisects ∠ A
(see Fig. 7.16). Show that Δ ABC Δ ABD.
What can you say about BC and BD?

**
Question 2 **. ABCD is a quadrilateral in which AD = BC and
∠ DAB = ∠ CBA (see Fig. 7.17). Prove that

(i) Δ ABD Δ BAC

(ii) BD = AC

(iii) ∠ ABD = ∠ BAC.

**Question 3**. AD and BC are equal perpendiculars to a line
segment AB (see Fig. 7.18). Show that CD bisects
AB.

**Question 4**. l and m are two parallel lines intersected by
another pair of parallel lines p and q
(see Fig. 7.19). Show that Δ ABC Δ CDA.

**Question 5**. line l is the bisector of an angle ∠ A and B is any
point on l. BP and BQ are perpendiculars from B
to the arms of ∠ A (see Fig. 7.20). Show that:

(i) Δ APB Δ AQB

(ii) BP = BQ or B is equidistant from the arms
of ∠ A.

**Question 6**. In Fig. 7.21, AC = AE, AB = AD and
∠ BAD = ∠ EAC. Show that BC = DE.

**
Question 7** . AB is a line segment and P is its mid-point. D and
E are points on the same side of AB such that
∠ BAD = ∠ ABE and ∠ EPA = ∠ DPB
(see Fig. 7.22). Show that

(i) Δ DAP Δ EBP

(ii) AD = BE

**Question 8**. In right triangle ABC, right angled at C, M is
the mid-point of hypotenuse AB. C is joined
to M and produced to a point D such that
DM = CM. Point D is joined to point B
(see Fig. 7.23). Show that:

(i) Δ AMC Δ BMD

(ii) ∠ DBC is a right angle.

(iii) Δ DBC Δ ACB

(iv) CM =
1
2 AB

EXERCISE 7.2

**Question 1**. In an isosceles triangle ABC, with AB = AC, the bisectors of ∠ B and ∠ C intersect
each other at O. Join A to O. Show that :

(i) OB = OC

(ii) AO bisects ∠ A

**
Question 2**. In Δ ABC, AD is the perpendicular bisector of BC
(see Fig. 7.30). Show that Δ ABC is an isosceles
triangle in which AB = AC.

**
Question 3**. ABC is an isosceles triangle in which altitudes
BE and CF are drawn to equal sides AC and AB
respectively (see Fig. 7.31). Show that these
altitudes are equal.

**Question 4**. ABC is a triangle in which altitudes BE and CF to
sides AC and AB are equal (see Fig. 7.32). Show
that

(i) Δ ABE Δ ACF

(ii) AB = AC, i.e., ABC is an isosceles triangle.

**
Question 5**. ABC and DBC are two isosceles triangles on the
same base BC (see Fig. 7.33). Show that
∠ ABD = ∠ ACD.

**
Question 6**. ΔABC is an isosceles triangle in which AB = AC.
Side BA is produced to D such that AD = AB
(see Fig. 7.34). Show that ∠ BCD is a right angle.
7. ABC is a right angled triangle in which ∠ A = 90°
and AB = AC. Find ∠ B and ∠ C.

**Question 7**. Show that the angles of an equilateral triangle
are 60° each.

EXERCISE 7.3

**Question 1**. Δ ABC and Δ DBC are two isosceles triangles on
the same base BC and vertices A and D are on the
same side of BC (see Fig. 7.39). If AD is extended
to intersect BC at P, show that

(i) Δ ABD Δ ACD

(ii) Δ ABP Δ ACP

(iii) AP bisects ∠ A as well as ∠ D.

(iv) AP is the perpendicular bisector of BC.

**
Question 2**. AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that

(i) AD bisects BC

(ii) AD bisects ∠ A.

**
Question 3**. Two sides AB and BC and median AM
of one triangle ABC are respectively
equal to sides PQ and QR and median
PN of Δ PQR (see Fig. 7.40). Show that:

(i) Δ ABM Δ PQN

(ii) Δ ABC Δ PQR

**
Question **4. BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence
rule, prove that the triangle ABC is isosceles.

**
Question 5**. ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC to show that
∠ B = ∠ C.

EXERCISE 7.4

**
Question 1**. Show that in a right angled triangle, the
hypotenuse is the longest side.

**
Question 2**. In Fig. 7.48, sides AB and AC of Δ ABC are
extended to points P and Q respectively. Also,
∠ PBC < ∠ QCB. Show that AC > AB.
3. In Fig. 7.49, ∠ B < ∠ A and ∠ C < ∠ D. Show that
AD < BC.

**
Question 3**. AB and CD are respectively the smallest and
longest sides of a quadrilateral ABCD
(see Fig. 7.50). Show that ∠ A > ∠ C and
∠ B > ∠ D.

**
Question 4**. In Fig 7.51, PR > PQ and PS bisects ∠ QPR. Prove
that ∠ PSR > ∠ PSQ.

EXERCISE 7.5

**Question 1** . ABC is a triangle. Locate a point in the interior of Δ ABC which is equidistant from all
the vertices of Δ ABC.

**Question 2**. In a triangle locate a point in its interior which is equidistant from all the sides of the
triangle.

**Question 3**. In a huge park, people are concentrated at three
points (see Fig. 7.52):

A : where there are different slides and swings
for children,

B : near which a man-made lake is situated,

C : which is near to a large parking and exit.
Where should an icecream parlour be set up so
that maximum number of persons can approach
it?
(Hint : The parlour should be equidistant from A, B and C)

**Question 4**. Complete the hexagonal and star shaped Rangolies [see Fig. 7.53(i) and (ii)] by filling
them with as many equilateral triangles of side 1 cm as you can. Count the number of
triangles in each case. Which has more triangles?

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- Chapter 3 Coordinate Geometry
- Chapter 4 Linear Equations In Two Variables
- Chapter 7 Triangles
- Chapter 8 Quadrilaterals
- Chapter 9 Areas of Parallelograms and Triangles
- Chapter 10 Circles
- Chapter 11 Constructions
- Chapter 12 Heron’s Formula
- Chapter 13 Surface Areas and Volumes
- Chapter 14 Statistics
- Chapter 15 Probability

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