NCERT Solutions for Class 9 Maths Chapter 11 - Constructions

Learn to draw triangles and bisectors of line segments with TopperLearning’s NCERT Solutions for CBSE Class 9 Mathematics Chapter 11 Constructions. Get step-by-step instructions to construct angles, bisectors and line segments. Revise the chapter solutions to understand how to give an appropriate justification for your construction.

Practise drawing arcs, triangles etc. according to the given data using the CBSE Class 9 Maths textbook solutions created by our subject experts. You can access these solutions any time at our online learning portal to revisit the topic of constructions and thus do well in your Maths exam.

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Chapter 11 - Constructions Exercise Ex. 11.1

Solution 1
Following are the steps of construction:

(i) Take the given ray PQ. Draw an arc of some radius taking point P as its centre, which intersect PQ at R.
 
(ii) Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.

(iii) Taking S as centre and with the same radius as before, drawn an arc intersecting the arc at T (see figure)

(iv) Taking S and T as centre, draw arc of same radius to intersect each other at U.

(v) Join PU, which is the required ray making 90o with given ray PQ.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
Justification of Construction:

We can justify the construction, if we can prove Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsUPQ = 90o.
For this let us join PS and PT
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
 
We have Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsSPQ = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsTPS = 60o. In (iii) and (iv) steps of this construction, we have drawn PU as the bisector of Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsTPS.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
Solution 2
The steps of construction are as follows:

(i)Take the given ray PQ. Draw an arc of some radius taking point P as its centre, which intersect PQ at R.
 
(ii)Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc  at S.
 
(iii)Taking S as centre and with the same radius as before, drawn an arc intersecting the arc at T (see figure)

(iv)Taking S and T as centre draw arc of same radius to intersect each other at U.
 
(v) Join PU. Let it intersect arc at point V.

(vi) Now from R and V draw arcs with other at W with radius more than Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions RV to intersect each other. PW is the required ray making 45o with PQ.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
 
Justification of Construction:

To justify the construction, we have to prove Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsWPQ = 45o.     Join PS and PT
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
We have Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsSPQ = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsTPS = 60o. In (iii) and (iv) steps of this construction, we have drawn PU as the bisector of Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsTPS.
Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsNcert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsUPS = Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions  Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsTPS= Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
Now, Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsUPQ = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsSPQ + Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsUPS
                  = 60o + 30o
                  = 90o
    In step (vi) of this construction, we constructed PW as the bisector of Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsUPQ
Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsNcert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsWPQ = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsUPQ = Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions  = 45o
 
Solution 3
(i) 30o
 
The steps of construction are as follows:

Step I: Draw the given ray PQ. Now taking P as centre and with some radius, draw an arc of a circle which intersects PQ at R.
 
Step II: Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at point S.
 
Step III: Now taking R and S as centre and with radius more than Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions RS draw arcs to intersect each other at T. Join PT which is the required ray making 30o with the given ray PQ.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
(ii) 22Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
          The steps of construction are as follows:
    
(i) Take the given ray PQ. Draw an arc of some radius, taking point P as its centre, which intersect PQ at R.

(ii) Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.
 
(iii) Taking S as centre and with the same radius as before, drawn an arc intersecting the arc at T (see figure)
 
(iv) Taking S and T as centre draw arc of same radius to intersect each other at U.

(v) Join PU. Let it intersect arc at point V.
 
(vi) Now from R and V draw arcs with radius more than  RV to intersect each Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions other at W. Join PW.
            
(vii) Let it intersects the arc at X. Taking X and R as centre and radius more than Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions RX draw arcs to intersect each other at Y. Joint PY which is the required ray making 22Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions  with the given ray PQ.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
 
(iii) 150
 
The steps of construction are as follows:

Step I: Draw the given ray PQ. Now taking P as centre and with some radius, draw an arc of a circle which intersects PQ at R.

Step II: Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at point S.  
 
Step III: Now taking R and S as centre and with radius more than Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions RS draw arcs to intersect each other at T. Join PT

Step IV: Let is intersects the arc at U. Now taking U and R as centre and with Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions radius more than RU draw arc to intersect each other at V. Join PV which is the required ray making 15o with given ray PQ.
 
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
Solution 4
(A) 75o
The steps of construction are as follows:

(i)  Take the given ray PQ. Draw an arc of some radius taking point P as its centre, which intersect PQ at R.

(ii) Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at  S.
 
(iii) Taking S as centre and with the same radius as before, drawn an arc intersecting the arc at T (see figure)

(iv) Taking S and T as centre draw arc of same radius to intersect each other at U.
 
(v)  Join PU. Let it intersects the arc at V. Now taking S and V as centre draw arcs with radius more thanNcert Solutions Cbse Class 9 Mathematics Chapter - Constructions   SV. Let those intersect each other at W. Join PW, which is the required ray making 75o with the given             ray PQ.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
 
Now, we can measure the angle so formed with the help of a protractor. It comes to be 75o.

(B) 105o

The steps of construction are as follows:

(i) Take the given ray PQ. Draw an arc of some radius taking point P as its centre, which intersect PQ at R.

(ii) Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.

(iii) Taking S as centre and with the same radius as before, drawn an arc intersecting the arc at T (see figure)

(iv) Taking S and T as centre draw arc of same radius to intersect each other at U.

(v) Join PU. Let it intersects the arc at V. Now taking T and V as centre draw arcs with radius more than Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions TV. Let these arcs intersect each other at W. Join PW, which is the required ray making 105o with the given ray PQ.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
 
Now, we can measure the angle so formed with the help of a protractor. It comes to be 105o.
                                                                                       
(C) 135o
The steps of construction are as follows:

(i) Take the given ray PQ. Extend PQ on opposite side of Q. Draw a semicircle of some radius taking point P as its centre, which intersect PQ at R and W.

(ii) Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.

(iii)Taking S as centre and with the same radius as before, drawn an arc intersecting the arc at T (see figure)

(iv) Taking S and T as centre, draw arc of same radius to intersect each other at U.

(v) Join PU. Let it intersect the arc at V. Now taking V and W as centre and with radius more than Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions  VW draw arcs to intersect each other at X. Join PX which is the required ray making 135o with the given line PQ.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
 
Now, we can measure the angle so formed with the help of a protractor. It comes to be 135o.                                    

Solution 5
We know that all sides of an equilateral triangle are equal. So, all sides of this equilateral triangle will be 5 cm.
Also, each angle of an equilateral triangle is 60.
    
The steps of construction are as follows:

Step I: Draw a line segment AB of 5 cm length. Draw an arc of some radius, while taking A as its centre. Let it intersect AB at P.    

Step II: Now taking P as centre draw an arc to intersect the previous arc at E. Join AE.

Step III: Taking A as centre draw an arc of 5 cm radius, which intersects extended line segment AE at C. Join AC and BC. Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsABC is the required equilateral triangle of side 5 cm.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
 
Justification of Construction:
To justify the construction, we have to prove that ABC is an equilateral triangle i.e.AB = BC = AC = 5 cm and Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsA = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsB = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsC = 60o.
 
Now, in Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsABC, we have AC = AB = 5 cm and Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsA = 60o
Since, AC = AB, we have
Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsB = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsC                         (angles opposite to equal sides of a triangle)
Now, in Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsABC
Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsNcert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsA + Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsB + Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsC = 180o    (angle sum property of a triangle)
 Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions60oNcert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsC + Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsC = 180o
 Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions60o + 2 Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsC = 180o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions 2 Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsC = 180o - 60o = 120o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsC = 60o
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
 
Now, we have Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsA = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsB = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsC = 60o              ... (1)
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsNcert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsA = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsB and Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsA = Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsC
Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsBC = AC and BC = AB          (sides opposite to equal angles of a triangle)
Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsAB = BC = AC = 5 cm                                  ... (2)
Equations (1) and (2) show that the Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsABC is an equilateral triangle.

Chapter 11 - Constructions Exercise Ex. 11.2

Solution 1
The steps of construction for the required triangles are as follows:
Step I: Draw a line segment BC of 7 cm. At point B draw an angle of 75o sayNcert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsXBC.  
Step II: Cut a line segment BD = 13 cm (that is equal to AB + AC) from the ray BX.  
Step III: Join DC and make an angle DCY equal to Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsBDC  
Step IV: Let CY intersects BX at A. Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsABC is the required triangle.   Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
 
Solution 2
The steps of construction for the required triangles are as follows:
Step I: Draw the line segment BC = 8 cm and at point B make an angle of 45o say Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsXBC.     
Step II: Cut the line segment BD = 3.5 cm (equal to AB - AC) on ray BX.

Step III: Join DC and draw the perpendicular bisector PQ of DC.

Step IV: Let it intersect BX at point A. Join AC. Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsABC is the required triangle.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
Solution 3
The steps of construction for the required triangles are as follows:
Step I: Draw line segment QR of 6 cm. At point Q draw an angle of 60o say Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsXQR.
Step II: Cut a line segment QS of 2 cm from the line segment QT extended an opposite side of line segment XQ. (As PR> PQ and PR - PQ = 2cm). Join SR.
Step III: Draw perpendicular bisector AB of line segment SR. Let it intersect QX at point P. Join PQ, PR. Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsPQR is the required triangle.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
Solution 4
The steps of construction for the required triangles are as follows:
 
Step I: Draw a line segment AB of 11 cm.(As XY + YZ + ZX = 11 cm)
Step II: Construct an angle Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsPAB of 30o at point A and an angle Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsQBA of 90o at point B.
Step III: Bisect Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsPAB and Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsQBA. Let these bisectors intersect each other at point X.
Step IV: Draw perpendicular bisector ST of AX and UV of BX. Step V: Let ST intersects AB at Y and UV intersects AB at Z.Join XY, XZ.Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsXYZ is the required triangle.
 
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions
Solution 5
The steps of construction for the required triangles are as follows:
Step I: Draw line segment AB of 12 cm. Draw a ray AX making 90o with AB.
 
Step II: Cut a line segment AD of 18 cm. (As sum of other two side is 18) from ray AX.
 
Step III: Join DB and make an angle DBY equal to ADB.
 
Step IV: Let BY intersects AX at C. Join AC, BC.Ncert Solutions Cbse Class 9 Mathematics Chapter - ConstructionsABC is the required triangle.
Ncert Solutions Cbse Class 9 Mathematics Chapter - Constructions