Q.1 Construct an angle of 90º at the initial point of a given ray and justify the construction.
Sol. Steps of construction :
1. Draw a ray OA.
2. With its initial point O as centre and any radius, draw an arc CDE, cutting OA at C.
3. With centre C and same radius (as in step 2), draw an arc, cutting the arc CDE at D.
4. With D as centre and the same radius, draw an arc cutting the arc CDE at E.
5. With D and E as centres, and any convenient radius , draw two arcs intersecting at P.
6. Join OP. Then
Justification : -
By construction , OC = CD = OD
Therefore OCD is an equilateral triangle. So,
Again OD = DE = EO
Therefore ODE is also an equilateral triangle. So
Since OP bisects .
Now,
Q.2 Construct an angle of 45º at the initial point of a given ray and justify the construction.
Sol.
Steps of Construction :
1. Draw a ray OA.
2. With O as centre and any suitable radius draw an arc cutting OA at B.
3. With B as centre and same radius cut the previous drawn arc at C and then with C as centre and same radius cut the arc at D.
4. With C as centre and radius more than half CD draw an arc.
5. With D as centre and same radius draw another arc to cut the previous arc at E.
6. Join OE. Then
7. Draw the bisector OF of
Justification :
By construction and OF is the bisector of
Therefore,
Q.3 Construct the angles of the following measurements :
(i) 30º (ii) (iii) 15º
Sol. (i) Steps of Construction :
1. Draw a ray OA.
2. With its initial point O as centre and any radius, draw an arc,cutting OA at C.
3. With centre C and Same radius (as in step 2). Draw an arc,cutting the arc of step 2 in D.
4. With C and D as centres, and any convenient radius ,draw two arcs intersecting at B.
5. Join OB. Then
(ii) Steps of Construction :
1. Draw an angle AOB = 90º
2. Draw the bisector OC of
3. Bisect , such that
Thus
(iii) Steps of Construction :
1. Construct an
2. Bisect so that .
3. Bisect , so that
Thus
Q.4 Construct the following angles and verify by measuring them by a protractor :
(i) 75º (ii) 105º (iii) 135º
Sol. (i) Steps of Construction :
1. Draw a ray OA.
2. Construct
3. Construct
4. Bisect so that
So, we obtain
= 60º + 15º = 75º
Verification :
On measuring , with the protractor, we find
(ii) Steps of Construction :
1. Draw a line segment XY.
2. Construct and , so that
= 120º – 90º
= 30º
3. Bisect angle SYT, by drawing its bisector YZ.
Then is the required angle of 105º
1. Draw
Then
2. Draw the bisector of .
Q.5 Construct an equilateral triangle, given its side and justify the construction.
Sol.
Let us draw an equilateral triangle of side 4.6 cm (say).
Steps of Construction :
1. Draw BC = 4.6 cm
2. With B and C as centres and radii equal to BC = 4.6 cm, draw two arcs on the same side of BC, intersecting each-other at A.
3. Join AB and AC.
Then, ABC is the required equilateral triangle.
Justification : Since by construction :
AB = BC = CA = 4.6 cm
Therefore ABC is an equilateral triangle.
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