NCERT Solutions for Class 9 Maths Chapter 7 Triangles

NCERT solutions for class 9 maths chapter 7 triangles banner image
author_logo

Written by Team Trustudies
Updated at 2021-02-14


NCERT solutions for class 9 Maths Chapter 7 Triangles Exercise 7.1

Q1 ) In quadrilateral ACBD, AC = AD and AB bisects ?A (see figure). Show that ?ABC ? ?ABD. What can you say about BC and BD ?
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?ABC and ?ABD, we have,

AC = AD ....(Given)

As, AB bisects ?A,

Therefore, ?CAB = ?DAB

Also, AB is a common side,

So, we can say,
?ABC ? ?ABD ...(by SAS test of congruency)

Also, BC = BD ...(By CPCT)
Hence, proved.

Q2 ) ABCD is a quadrilateral in which AD = BC and ?DAB = ?CBA (see figure). Prove that
(i) ?ABD ? ?BAC
(ii) BD = AC
(iii) ?ABD = ?BAC
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?ABC and ?BAC, we have

AD = BC ...(Given)
Also,
?DAB = ?CBA ...(Given)

And AB is a common side.

Therefore, ?ABD ? ?BAC ...(SAS congruency test)

Hence, BD = AC ...(By CPCT)

and ?ABD = ?BAC ...(By CPCT)
Hence, proved.

Q3 ) AD and BC are equal perpendiculars to a line segment AB (see figure). Show that CD bisects AB.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?AOD and ?BOC, we have,

Now, ?AOD = ?BOC ...(vertically opposite angles)

Also, ?DAO = ?CBO = 90?

and BD = BC ...(shown in the figure)

Therefore, ?AOD ? ?BOC ...(SAS congruency test)

Hence, OA = OB ...(By CPCT)

Thus, we can say that, O is the mid-point of AB.

So, CD bisects AB.
Hence, proved

Q4 ) l and m are two parallel lines intersected by another pair of parallel lines p and q (see figure). Show that ?ABC ? ?CDA.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

image

From figure, we have,

?1 = ?2 (Vertically opposite angles) ...(i)

?1 = ?6 (Corresponding angles) ...(ii)
?6 = ?4 (Corresponding angles) ...(iii)

From Equations, (i), (ii) and (iii), we have

?1 = ?4 and ?2 = ?6 ...(iv)

In ?ABC and ?CDA, we have

?4 = ?2 ...(from (iii) and (iv))
?5 = ?3 ...(Alternate angles)

and AC is a common side.

Therefore, we get,
?ABC ? ?CDA ...(By SAS congruency test)
Hence, proved.

Q5 ) Line l is the bisector of a ?A and ?B is any point on l. BP and BQ are perpendiculars from B to the arms of ?A (see figure). show that
(i) ?APB ? ?AQB
(ii) BP = BQ or B is equidistant from the arms of ?A.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?APB ? ?AQB, we have

?APB = ?AQB = 90?
?PAB = ?QAB ...(AB bisects ?PAQ)

AB is the common side.
? ? {APB}\) ? ?AQB ...(By AAS congruency test)

Also, BP = BQ ...(By CPCT)

Thus, we can say that, B is equidistant from the arms of ?A.
Hence, proved.

Q6 ) In figure, AC = AE, AB = AD and ?BAD = ?EAC. Show that BC = DE.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?ABC and ?ADE , we have,

AB = AD ...(Given)
?BAD = ?EAC ...(i)(Given)
On adding, ?DAC on both sides in Eq. (i), we get,

? ?BAD + ?DAC = ?EAC + ?DAC
? ?BAC = ?DAE

and also, AC = AE ...(Given)

? ?ABC ? ?ADE ...(By AAS congruency test)

Thus, BC = DE ...(By CPCT)

Hence, proved.

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 figure). Show that
(i) ?DAP ? ?EBP
(ii) AD = BE
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

We have,
AP = BP ...(i)(Since, P is the mid-point of AB)

?EPA = ?DPB ...(ii)(Given)
?BAD = ?ABE ...(iii) (Given)
On adding, ?EPD on both sides in Equation (ii),
we have,
? ?EPA + ?EPD = ?DPB + ?EPD
? ?DPA = ?EPB ...(iv)

Now, In ?DAP and ?EBP We have,
? ?DPA = ?EPB ...(from (iv)),
? ?DAP = ?EBP ...(Given)
and AP = BP ...(From Eq. (i))

? ?DAP ? ?EBP ...(By ASA congruency test)

Thus, AD = BE ...(By CPCT)

Hence, proved.

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 figure).
image
Show that
(i) ?AMC ? ?BMD
(ii) ?DBC is a right angle
(iii) ?DBC ? ?ACB
(iv)CM = (1/2) AB



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

Given :
?ACB in which ?C = 90? and M is the mid-point of AB.

To prove :
(i) ?AMC ? ?BMD
(ii) ?DBC is a right angle
(iii) ?DBC ? ?ACB
(iv)CM = (12 ) AB

image

Construction : Produce CM to D, such that CM = MD. Join DB.

Proof : In ?AMC and ?BMD, we have

AM = BM ...(M is the mid-point of AB)
CM = DM ...(Given)
and ?AMC = ?BMD ...(Vertically opposite angles)
? ?AMC ? ?BMD(By SAS congruency test)
Hence, part(i) is proved.


Also, AC = DB ...(By CPCT)

?1 = ?2 ...(Alternate angles) and (by CPCT)

? BD || CA and BC is transversal.

? ?ACB + ?DBC = 180?
But, ?ACB = 90? ...(given)
? 90? + ?DBC = 180?
? ?DBC = 180? - 90?
? ?DBC = 90?

Hence, part(ii) is proved, too.


Now, considering ?DBC and ?ACB, we have,

AC = DB ...(from part(i))

Side BC is common.

and ?DBC = ?ACB = 90?

Therefore, ?DBC ? ?ACB ...(SSA Congruency theorem)
Hence, now, part(iii) is proved, too.


Now, DC = AB ...(By CPCT)

Multipling both sides by 12 , we get,
(12 ) DC = 12 ) AB

Now, as we know, CM = (12 ) DC
Therefore, CM = (12 ) AB
Hence, part (iv) is proved.

NCERT solutions for class 9 Maths Chapter 7 Triangles Exercise 7.2

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



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?ABC, we have

AB = AC ...(Given)
Also, ?B = ?C ...(Since corresponding angles of equal sides are equal)

Multipling both sides by 12 , we get,
(12 ) ?B = (12 ) ?C
? ?OBC = ?OCB

Also, it is given that, OB and OC are bisectors of ?B and ?C, respectively,

Therefore, ?OBA and ?OCA
Therefore, OB = OC
...(? corresponding sides of equal angles are equal)

Hence, part (i) is proved.

image

In ?ABO and ?ACD, we have,

AB = AC ...(Given)
?OBA = ?OCA ...(from part(i))
OB = OC ...(proved earlier)

Therefore, ?ABO ? ?ACO ...(By SAS congruency test)

Thus, ?BAO = ?CAO ...(By CPCT)

Hence, AO bisects ?A is proved.

Q2 ) In the ?ABC, AD is the perpendicular bisector of BC (see figure).
Show that ?ABC is an isosceles triangle in which AB = AC.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?ABC and ?ACD, we have,

DB = DC ...(given)
?ADB = ?ADC ...(Since, AD is the perpendicular bisector of BC)

and AD is the Common side.

Therefore, ?ABD ? ?ACD ...(By SAS congruency test)

Therefore, AB = AC ...(By CPCT)

Hence, ?ABC is an isosceles triangle.
Hence, proved.

Q3 ) ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see figure).
Show that these altitudes are equal.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?ABE and ?ACF , we have,

?AEB = ?AFC ...(BE and CF are perpendiculars drawn to sides AC and AB respectively)

Also, ?A is common angle.
and AB = AC ...(Given)

Therefore, ?ABE ? ?ACF ...(By AAS congruency test)

Therefore, AB = AC ...(By CPCT)

Hence, BE = CF ...(By CPCT)
Hence, proved.

Q4 ) ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see figure). Show that
(i) ?ABE ? ?ACF
(ii) AB = AC i.e., ABC is an isosceles triangle.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?ABE and ?ACF , we have,

?AEB = ?AFC(?BEandCFareperpendiculartosidesACandAB)\(?BAE = ?CAF ...(?\(?A is the Common angle)

and BE = CF ...(Given)

? ?ABE ? ?ACF ...(By AAS Congruency test)

Thus, AB = AC ...(By CPCT)

? ?ABC is an isosceles triangle.
Hence, proved.

Q5 ) ABC and DBC are isosceles triangles on the same base BC (see figure). Show that ?ABD = ?ACD.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?ABC, we have,

AB = AC ...(? ?ABC is an isosceles triangle)

? ?ABC = ?ACB ...(i)(? angles opposite to equal sides are equal)

Similarly, in ?DBC, we have,

BD = CD ...(? ?DBC too, is an isosceles triangle)

? ?DBC = ?DCB ...(ii) (? angles opposite to equal sides are equal)
Now, On adding, Equations (i) and (ii), we get,

??ABC+?DBC=?ACB+?DCB
? ?ABD = ?ACD
Hence, proved.

Q6 ) ?ABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see figure).
Show that ?BCD is a right angle.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?ABC, we have,

AB = AC ...(i)(Given)
?ACB = ?ABC ...(ii)(? angles opposite to equal sides are equal)
Also, AB = AD ...(iii)(Given)

From (i) and (iii), AC = AD

Now, in ?ADC , we have,

AD = AC ...(proved earlier)
?ACD = ?ADC ...(? angles opposite to equal sides are equal)

Also, ?ACD = ?BDC ...(iv)(??ADC=?BDC)

On adding Equations (ii) and (iv), we get,

??ACB+?ACD=?ABC+?BDC
? ?BCD = ?ABC + ?BDC

Adding ?BCD on both sides, we have,

??BCD+?BCD=?ABC+?BDC+?BCD
? 2 ?BCD = 180? ...(? sum of all angles of a triangle is 180?)

Therefore, ?BCD = 90?
Hence, proved.

Q7 ) ABC is a right angled triangle in which ?A = 90? and AB = AC, find ?B and ?C.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

In ?ABC, we have,

AB = AC ...(Given)
?B = ?C ...(i)(? angles opposite to equal sides are equal)

Now, we know that,

?A + ?B + ?C = 180?
? 90? + ?B + ?C = 180? ...(given)
? 90? + ?B + ?B = 180? ...(from(i))
? 2 ?B = 180? - 90?
? 2 ?B = 90?
? ?B = 45?
? ?C = 45?, too.

Q8 ) Show that the angles of an equilateral triangle are 60? each.



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

Let ?ABC be an equilateral triangle, such that

AB = BC = CA (by property)

image

Now, we have,
AB = AC
?B = ?C ...(i)(? angles opposite to equal sides are equal)

Similarly,
CB = CA

? ?A = ?B ...(ii)(? angles opposite to equal sides are equal)

?A + ?B + ?C = 180? ...(iii)(? the sums of all angles of a triangle are 180?)

From Equations (i),(ii) and (iii), we have,
?A + ?A + ?A = 180?
? 3 ?A = 180?
? ?A = 60?
??A=?B=?C=60?
Hence, proved.

NCERT solutions for class 9 Maths Chapter 7 Triangles Exercise 7.3

Q1 ) ?ABC and ?DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see figure). If AD is extended to intersect BC at P, show that
(i) ?ABD ? ?ACD
(ii)?ABP ? ?ACP
(iii) AP bisects ?A as well as ?D
(iv) AP is the perpendicular bisector of BC.
image



NCERT Solutions for Class 9 Maths Chapter 7 Triangles


Answer :

Given: ?ABC and ?DBC are two isosceles triangles having same base BC, such that AB = AC and BD = CD.
image
Proof:
In ?ABD and ?ACD, we have,

AB = AC ...(Given)
BD = CD ...(Given)
and AD is Common side.

Therefore, ?ABD ? ?ACD ...(By SSS Congruency test)
Hence, part (i) is proved.


In ?ABP and ?ACP, we have,

AB = AC ...(Given)
Also, ?a = ?b ...(? ?ABD ? ?ACD)
and AP is the common side.

Therefore, ?ABP