NCERT Solutions for Class 9 Maths Chapter 11 Constructions Ex 11.1
Question 1:
Construct an angle of 90° at the initial point of a given ray and justify the construction.
Solution 1:
The below given steps will be followed to construct an angle of 90°.
(i)Take the given ray PQ. Draw an arc of some radius taking point P as its centre, whichintersects PQ at R.
(ii)Taking R as centre and with the same radius as before, draw an arc intersecting thepreviously drawn arc at S.
(iii)Taking S as centre and with the same radius as before, draw an arc intersecting the arc at T(see figure).
(iv)Taking S and T as centre, draw an arc of same radius to intersect each other at U.
(v)Join PU, which is the required ray making 90° with the given ray PQ.
Question 2:
Construct an angle of 45° at the initial point of a given ray and justify the construction.
Solution 2:
The below given steps will be followed to construct an angle of 45°.
(i)Take the given ray PQ. Draw an arc of some radius taking point P as its centre, which intersects 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, draw an arc intersecting the arc at T(see figure).
(iv)Taking S and T as centre, draw an arc of same radius to intersect each other at U.
(v)Join PU. Let it intersect the arc at point V.
(vi)From R and V, draw arcs with radius more than 1/2 RV to intersect each other at W. Join PW.
PW is the required ray making 45° with PQ.
Question 3:
Construct the angles of the following measurements:
(i) 30° (ii)22.5° (iii) 15°
Solution 3 :
(i)30°
The below given steps will be followed to construct an angle of 30°.
Step I: Draw the given ray PQ. Taking P as centre and with some radius, draw an arc of a circlewhich 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: Taking R and S as centre and with radius more than 1/2 RS, draw arcs to intersect each other at T. Join PT which is the required ray making 30° with the given ray PQ.
(ii)22.5°
The below given steps will be followed to construct an angle of 22.5°: -
(1)Take the given ray PQ. Draw an arc of some radius, taking point P as its centre, which intersects PQ at R.
(2)Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.
(3)Taking S as centre and with the same radius as before, draw an arc intersecting the arc at T(see figure).
(4)Taking S and T as centre, draw an arc of same radius to intersect each other at U.
(5)Join PU. Let it intersect the arc at point V.
(6)From R and V, draw arcs with radius more than 1/2 RV to intersect each other at W. Join PW.
(7)Let it intersect the arc at X. Taking X and R as centre and radius more than 1/2 RX, draw arcs to intersect each other at Y.
Joint PY which is the required ray making 22.5° with the given ray PQ.
(iii)15°
The below given steps will be followed to construct an angle of 15°.
Step I: Draw the given ray PQ. 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: Taking R and S as centre and with radius more than 1/2 RS, draw arcs to intersect each other at T. Join PT.
Step IV: Let it intersect the arc at U. Taking U and R as centre and with radius more than 1/2 RU, draw an arc to intersect each other at V.
Join PV which is the required ray making 15° with the given ray PQ.
Question 4:
Construct the following angles and verify by measuring them by a protractor:
(i)75° (ii) 105° (iii) 135°
Solution 4:
(i)75°
The below given steps will be followed to construct an angle of 75°.
(1)Take the given ray PQ. Draw an arc of some radius taking point P as its centre, which intersects PQ at R.
(2)Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.
(3)Taking S as centre and with the same radius as before, draw an arc intersecting the arc at T(see figure).
(4)Taking S and T as centre, draw an arc of same radius to intersect each other at U.
(5)Join PU. Let it intersect the arc at V. Taking S and V as centre, draw arcs with radius more than 1/2 SV. Let those intersect each other at W.
Join PW which is the required ray making 75° with the given ray PQ.
The angle so formed can be measured with the help of a protractor. It comes to be 75º.
(ii)105°
The below given steps will be followed to construct an angle of 105°.
(1)Take the given ray PQ. Draw an arc of some radius taking point P as its centre, which intersects PQ at R.
(2)Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.
(3)Taking S as centre and with the same radius as before, draw an arc intersecting the arc at T(see figure).
(4)Taking S and T as centre, draw an arc of same radius to intersect each other at U.
(5)Join PU. Let it intersect the arc at V. Taking T and V as centre, draw arcs with radius more than 1/2 TV. Let these arcs intersect each other at W.
Join PW which is the required ray making 105° with the given ray PQ.
The angle so formed can be measured with the help of a protractor. It comes to be 105º.
(iii)135°
The below given steps will be followed to construct an angle of 135°.
(1)Take the given ray PQ. Extend PQ on the opposite side of Q. Draw a semi-circle of some radius taking point P as its centre, which intersects PQ at R and W.
(2)Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.
(3)Taking S as centre and with the same radius as before, draw an arc intersecting the arc at T(see figure).
(4)Taking S and T as centre, draw an arc of same radius to intersect each other at U.
(5) Join PU. Let it intersect the arc at V. Taking V and W as centre and with radius more than 1/2 VW,draw arcs to intersect each other at X.Join PX, which is the required ray making 135° with the given line PQ.
The angle so formed can be measured with the help of a protractor. It comes to be 135º.
Question 5:
Construct an equilateral triangle, given its side and justify the construction.
Solution 5:
Let us draw an equilateral triangle of side 5 cm. We know that all sides of an equilateral triangle are equal. Therefore, all sides of the equilateral triangle will be 5 cm. We also know that each angle of an equilateral triangle is 60º.
The below given steps will be followed to draw an equilateral triangle of 5 cm side.
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: 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.
Triangle ABC is the required equilateral triangle of side 5 cm.
