NCERT Solutions for Class 9 Maths Chapter 7 - Triangles

Chapter 7 - Triangles Exercise Ex. 7.1

Solution 1
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABD
AC = AD                                            (given)
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCAB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDAB                                 (given)
AB = AB                                             (common)
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
So, BC and BD are of equal length.

Solution 2
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABD and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAC
    AD = BC                                         (given)
   Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDAB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCBA                              (given)
    AB = BA                                         (common)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
And Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABD = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAC                          (by CPCT)
Solution 3
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBOC and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAOD
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBOC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAOD                                 (vertically opposite angles)
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCBO = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDAO                                 (each 90o)
BC = AD                                             (given)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Solution 4
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Solution 5
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Solution 6
Given that Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAD = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesEAC
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAD + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDAC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesEAC + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDAC
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDAE
Now in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAC and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDAE
AB = AD                                             (given)
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDAE                                 (proved above)
AC = AE                                             (given)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Solution 7
Given that Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesEPA = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDPB
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesEPA + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDPE = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDPB + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDPE
 Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesNcert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDPA = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesEPB
     Now in  Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDAP and Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles EBP
    Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDAP = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesEBP                               (given)
    AP = BP                                          (P is mid point of AB)
    Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDPA = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesEPB                              (from above)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Solution 8
 (i)  In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAMC and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBMD
AM = BM                                              (M is mid point of AB)
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAMC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBMD                                  (vertically opposite angles)
CM = DM                                             (given)
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
(ii) We have Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACM = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBDM
But Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACM and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBDM are alternate interior angles
Since alternate angles are equal.  
Hence, we can say that DB || AC
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDBC + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB = 180o                   (co-interior angles) Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDBC + 90o = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDBC + 90o = 1800 
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesNcert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDBC          =  90o
 
(iii) Now in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDBC and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB
DB = AC                                             (Already proved)
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDBC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB                                 (each 90o )
BC = CB                                             (Common)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
(iv) We have Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesDBC Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB
 Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles

Chapter 7 - Triangles Exercise Ex. 7.2

Solution 1
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
(i)    It is given that in triangle ABC, AC = AB             
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles  Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC     (angles opposite to equal sides of a triangle are equal)
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesNcert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesOBC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesOBC
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesOB = OC                  (sides opposite to equal angles of a triangle are also equal)
 
(ii) Now in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesOAB and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesOAC
    AO =AO                               (common)
   AB = AC                                (given)
   OB = OC                               (proved above)    
   So, Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesOAB Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesOAC         (by SSS congruence rule)
   Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAO = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCAO             (C.P.C.T.)
Solution 2
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesADC and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesADB
    AD = AD                                         (Common)
    Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesADC =Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesADB                              (each 90o)
    CD = BD                                        (AD is the perpendicular bisector of BC)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Solution 3
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAEB and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAFC
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAEB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAFC                                             (each 90o)
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA                                                     (common angle)
AB = AC                                                        (given)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Solution 4
(i) In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAEB and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAFC
    Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAEB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAFC                          (each 90)
    Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA                                      (common angle)
     BE = CF                                        (given)
     Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
(ii) We have already proved
    Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAEB Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAFC
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles AB = AC                                      (by CPCT)

Solution 5
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
Let us join AD
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABD and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACD
AB = AC                                    (Given)
BD = CD                                    (Given)
AD = AD                                    (Common side)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Solution 6
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC
AB = AC                                                (given)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC                               (angles opposite to equal sides of a triangle are also equal)
Now In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACD
AC = AD
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesADC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACD                              (angles opposite to equal sides of a triangle are also equal)
Now, in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBCD
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBCD + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesADC = 180o          (angle sum property of a triangle)
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesNcert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB +Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACD + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACD = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles 2(Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACD) = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles 2(Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBCD) = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBCD = 90o
Solution 7
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Given that
AB = AC
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB                      (angles opposite to equal sides are also equal)
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC,
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC = 180o     (angle sum property of a triangle)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles 90o + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles 90o + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles 2 Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB = 90o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB = 45
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Solution 8
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
Let us consider that ABC is an equilateral triangle.
So, AB = BC = AC
Now, AB = AC
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB         (angles opposite to equal sides of a triangle are equal)
    
We also have
AC = BC    
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA             (angles opposite to equal sides of a triangle are equal)
    
So, we have
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC
    Now, in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles 3Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA = 60o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC = 60o
Hence, in an equilateral triangle all interior angles are of 60o.

Chapter 7 - Triangles Exercise Ex. 7.3

Solution 1
(i)  In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABD and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACD
      AB = AC                                             (given)
      BD = CD                                            (given)
      AD = AD                                            (common)
      Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
(ii)  In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABP and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACP
       AB = AC                                            (given).
      Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAP = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCAP                                  [from equation (1)]
       AP = AP                                             (common)
       Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
(iii)   From equation (1)
        Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAP = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCAP            
        Hence, AP bisect Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA
        Now in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBDP and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCDP
        BD = CD                                            (given)
        DP = DP                                            (common)
        BP = CP                                            [from equation (2)]
        Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
(iv)   We have Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBDP Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCDP
       Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
       

         Now, Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBPD + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCPD = 180o             (linear pair angles)

         Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBPD + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBPD = 180o 

         2Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBPD = 180o                                    [from equation (4)]

        Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBPD = 90o                                                                    ...(5)

        From equations (2) and (5), we can say that AP is perpendicular  bisector of BC.

      

Solution 2
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
(i)   In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAD and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCAD
        Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesADB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesADC                                      (each 90o as AD is an altitude)
        AB = AC                                                  (given)
        AD = AD                                                  (common)
  Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
(ii)     Also by CPCT,
         Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBAD = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCAD
          Hence, AD bisects Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA.

(ii)              Also by CPCT,

          ÐBAD = ÐCAD

         Hence, AD bisects ÐA.

 

Solution 3
(i)  In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC, AM is median to BC
     Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBM = Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles BC
     In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesPQR, PN is median to QR
     Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles QN = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesQR
     But BC = QR
 
      Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
     Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles BN = QN                                                     ...(i)
      Now, in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABM and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesPQN
     AB = PQ                                                       (given)
     BM = QN                                                       [from equation (1)]
     AM = PN                                                        (given)
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesNcert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
 (ii)  Now in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesPQR

      AB = PQ                                                        (given)
     Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesPQR                                             [from equation (2)]
     BC = QR                                                        (given)
     Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesPQR                                  (by SAS congruence rule)  

Solution 4
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesIn Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBEC and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCFB
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBEC = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCFB                                              (each 90o )
BC = CB                                                         (common)
BE = CF                                                         (given)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles                                                 (Sides opposite to equal angles of a triangle are equal)
 
Hence, Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC is isosceles.
Solution 5
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAPB and Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAPC
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAPB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAPC                                             (each 90o)
AB =AC                                                          (given)
AP = AP                                                   
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
     (common)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB = Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC                                                (by using CPCT)

Chapter 7 - Triangles Exercise Ex. 7.4

Solution 1
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Let us consider a right angled triangle ABC, right angle at B.
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC = 180o            (angle sum property of a triangle)
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA + 90o + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC = 90o
Hence, the other two angles have to be acute (i.e. less than 90).
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
[In any triangle, the side opposite to the larger (greater) angle is longer]
So, AC is the largest side in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC.
But AC is the hypotenuse of Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC. Therefore, hypotenuse is the longest side in a right angled triangle.

Solution 2
In the given figure,
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesPBC = 180p            (linear pair)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC = 180o - Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesPBC             ... (1)
Also,
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesQCB = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB = 180o - Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesQCB                    ... (2)
As Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesPBC < Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesQCB
 Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles 180 - Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesPBC > 180o - Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesQCB.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC > Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesACB                [From equations (1) and (2)]
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles AC > AB                           (side opposite to larger angle is larger)
Solution 3
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesAOB
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB < Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles AO < BO     (side opposite to smaller angle is smaller)        ... (1)
Now in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesCOD
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC < Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesD
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles OD < OC     (side opposite to smaller angle is smaller)        ... (2)
On adding equations (1) and (2), we have
AO + OD < BO + OC
AD < BC

Solution 4
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Let us join AC.
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC
AB < BC           (AB is smallest side of quadrilateral ABCD)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
(1)
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesADC
AD < CD          (CD is the largest side of quadrilateral ABCD)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
(2)
On adding equations (1) and (2), we have
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles2 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles4 < Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles1 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles3
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC < Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesA > Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesC
Let us join BD.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABD
AB < AD            (AB is smallest side of quadrilateral ABCD)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
(3)
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesBDC
 BC < CD         (CD is the largest side of quadrilateral ABCD)
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
On adding equations (3) and (4), we have
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles8 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles7 < Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles5 + Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles6
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesD < Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesB > Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesD

Solution 5
  As PR > PQ
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
  PS is the bisector of Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesQPR
  Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
   Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Solution 6
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
Let us take a line l and from point P (i.e. not on line l) we have drawn two line segments PN and PM. Let PN be perpendicular to line l and PM is drawn at some other angle.
In Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesPNM
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesN = 90o
Now, Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesP + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesN + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesM = 180o    (Angle sum property of a triangle)    
Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesP + Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesM = 90o            
Clearly, Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesM is an acute angle
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles

Chapter 7 - Triangles Exercise Ex. 7.5

Solution 1
Circumcentre of a triangle is always equidistant from all the vertices of that triangle. Circumcentre is the point where perpendicular bisectors, of all the sides of triangles meet together.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
As here in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC we can find the circumcentre by drawing the perpendicular bisectors of sides AB, BC, and CA of this triangle. O is the point where these bisectors are meeting together. So O is point which is equidistant from all the vertices of Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC.
Solution 2
The point which is equidistant from all the sides of a triangle is incenter of triangle. Incentre of triangle is the intersection point of angle bisectors of interior angles of that triangle.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Here in Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC we can find the incentre of this triangle by drawing the angle bisectors of interior angles of this triangle. I is the point where these angle bisectors are intersecting each other. So, I is the point, equidistant from all the sides of Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC.
Solution 3
Ice-cream parlour should be set up at the circumcentre O of Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesABC.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
In this situation maximum number of persons can approach to it. We can find circumcentre O of this triangle by drawing perpendicular bisectors of the sides of this triangle.

Solution 4
We may observe that hexagonal shaped rangoly is having 6 equilateral triangles in it.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Area of Ncert Solutions Cbse Class 9 Mathematics Chapter - TrianglesOAB = Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles (side)2 = Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles (5)2
                          Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
    Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles

 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Triangles