# NCERT Solutions for Class 9 Maths Unit 7

## Triangles Class 9

### Unit 7 Triangles Exercise 7.1, 7.2, 7.3, 7.4, 7.5 Solutions

Triangle in mathematics, plane figure bounded by three straight lines, the sides, which intersect at three points called the vertices. Any one of the sides may be considered the base of the triangle. The perpendicular distance from a base to the opposite vertex is called an altitude. The area of a triangle is equal to one half the product of the base and the corresponding altitude. The line segment joining the midpoint of a side to the opposite vertex is called a median. All three altitudes of a triangle go through a single point, and all three medians go through a single (usually different) point.
In Euclidean geometry the sum of the angles of a triangle is equal to two right angles (180�). If all three angles of a triangle are equal, the triangle is called equilateral. An isosceles triangle has two equal angles. A scalene triangle is one in which all three angles are different. A right triangle has one right angle. In geometry it is shown that two triangles are congruent (i.e., are the same shape and size) if, in general, any three independent parts (sides or angles) of one are the same as the corresponding three parts of the other. The rules of congruency make it possible, in trigonometry, to compute the sides and the angles of a triangle when three of these values are known.
The triangle is the simplest of the polygons (i.e., it has the least possible number of sides). Since any polygon can be broken up into triangles by drawing various diagonals, a complete theory of the measurement of triangles provides a complete theory of the measurement of all polygons. In non-Euclidean geometries, the angles of a triangle are either less than two right angles (hyperbolic geometry) or more than two right angles (elliptic geometry).

### Exercise 7.1 : Solutions of Questions on Page Number : 118

Q1 :

In quadrilateral ACBD, AC = AD and AB bisects ∠ A (See the given figure). Show that ΔABC ≅ ΔABD. What can you say about BC and BD? In ΔABC and ΔABD,

∠ CAB = ∠ DAB (AB bisects ∠ A)

AB = AB (Common)

∴ ΔABC ≅ ΔABD (By SAS congruence rule)

∴ BC = BD (By CPCT)

Therefore, BC and BD are of equal lengths.

Q2 :

ABCD is a quadrilateral in which AD = BC and ∠ DAB = ∠ CBA (See the given figure). Prove that

(i) ΔABD ≅ ΔBAC

(ii) BD = AC

(iii) ∠ ABD = ∠ BAC. In ΔABD and ΔBAC,

∠ DAB = ∠ CBA (Given)

AB = BA (Common)

∴ ΔABD ≅ ΔBAC (By SAS congruence rule)

(ii). ∴ BD = AC (By CPCT)

(iii). And, ∠ ABD = ∠ BAC (By CPCT)

Q3 :

AD and BC are equal perpendiculars to a line segment AB (See the given figure). Show that CD bisects AB. Q4 :

l and m are two parallel lines intersected by another pair of parallel lines p and q (see the given figure). Show that ΔABC ≅ ΔCDA. Q5 :

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 the given figure). Show that:

(i) ΔAPB ≅ ΔAQB

(ii) BP = BQ or B is equidistant from the arms of ∠ A. Q6 :

In the given figure, AC = AE, AB = AD and ∠ BAD = ∠ EAC. Show that BC = DE. Q7 :

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 the given figure). Show that

(i) ΔDAP ≅ ΔEBP Q8 :

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 the given figure). Show that:

(i) ΔAMC ≅ ΔBMD

(ii) ∠DBC is a right angle.

(iii) ΔDBC ≅ ΔACB

(iv) CM = AB ### Exercise 7.2 : Solutions of Questions on Page Number : 123

Q1 :

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

Q2 :

In ΔABC, AD is the perpendicular bisector of BC (see the given figure). Show that ΔABC is an isosceles triangle in which AB = AC. Q3 :

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see the given figure). Show that these altitudes are equal. Q4 :

ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that

(i) ABE ≅ ACF

(ii) AB = AC, i.e., ABC is an isosceles triangle. Q5 :

ABC and DBC are two isosceles triangles on the same base BC (see the given figure). Show that ∠ ABD = ∠ ACD. Q6 :

ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see the given figure). Show that ∠ BCD is a right angle. Q7 :

ABC is a right angled triangle in which ∠ A = 90º and AB = AC. Find ∠ B and ∠ C.

Q8 :

Show that the angles of an equilateral triangle are 60º each.

### Exercise 7.3 : Solutions of Questions on Page Number : 128

Q1 :

ΔABC and ΔDBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see the given figure). If AD is extended to intersect BC at P, show that

(i) ΔABD ≅ ΔACD

(ii) ΔABP ≅ ΔACP

(iii) AP bisects ∠ A as well as ∠ D.

(iv) AP is the perpendicular bisector of BC. Q2 :

AD is an altitude of an isosceles triangles ABC in which AB = AC. Show that

Q3 :

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see the given figure). Show that:

(i) ΔABM ≅ ΔPQN

(ii) ΔABC ≅ ΔPQR Q4 :

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

Q5 :

ABC is an isosceles triangle with AB = AC. Drawn AP ⊥ BC to show that ∠ B = ∠ C.

### Exercise 7.4 : Solutions of Questions on Page Number : 132

Q1 :

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

Q2 :

In the given figure sides AB and AC of ΔABC are extended to points P and Q respectively. Also, ∠ PBC < ∠ QCB. Show that AC > AB. Q3 :

In the given figure, ∠ B < ∠ A and ∠ C < ∠ D. Show that AD < BC. Q4 :

AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD (see the given figure). Show that ∠ A > ∠ C and ∠ B > ∠ D. Q5 :

In the given figure, PR > PQ and PS bisects ∠ QPR. Prove that ∠ PSR >∠ PSQ. Q6 :

Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

### Exercise 7.5 : Solutions of Questions on Page Number : 133

Q1 :

ABC is a triangle. Locate a point in the interior of ΔABC which is equidistant from all the vertices of ΔABC.

Q2 :

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

Q3 :

In a huge park people are concentrated at three points (see the given figure) 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 ice-cream parlour be set up so that maximum number of persons can approach it?

(Hint: The parlor should be equidistant from A, B and C)

Q4 :

Complete the hexagonal and star shaped rangolies (see the given figures) 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? 