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NCERT solutions for class 9 maths chapter 11 Constructions
Excercise: 11.1
Q1 Construct an angle of 90 o at the initial point of a given ray and justify the construction.
Answer:
The steps of construction to follow:
Step 1: Draw a ray OP.
Then, take O as the centre and any radius draw an arc cutting OP at Q.
Step 2: Now, taking Q as the centre and with the same radius as before draw an arc cutting the
previous arc at R. Repeat the process with R to cut the previous arc at S.
Step 3: Take R and S as centre draw the arc of radius more than the half of RS and draw two
arcs intersecting at A. Then, join OA. Aakas
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Hence, .
Justification:
We need to justify,
So, join OR and OS and RQ. we obtain
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By construction OQ = OS = QR.
So, is an equilateral triangle. Similarly is an equilateral triangle.
So,
Now, that means .
Then, join AS and AR:
Now, in triangles OSA and ORA:
(common)
(Radii of same arcs)
(radii of the same arcs)
So,
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Therefore,
and
Hence, justified.
Q2 Construct an angle of 45 o at the initial point of a given ray and justify the construction.
Answer:
The steps of construction to follow:
Step 1: Draw a ray OY.
Then, take O as the centre and any radius, mark a point A on the arc ABC.
Step 2: Now, taking A as the centre and the same radius, mark a point B on the arc ABC.
Step 3: Take B as a centre and the same radius, mark a point C on the arc ABC.
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Step 4: Now, taking C and B as centre one by one, draw an arc from each centre intersecting
each other at a point X.
Step 5: X and O are joined and a ray making an angle with OY is formed.
Let the arc AC touches OX at E
Step 6: With A and E as centres, 2 arcs are marked intersecting each other at D and the bisector
of angle XOY is drawn.
Justification:
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By construction we have,
We constructed the bisector of as
Thus,
Q3 (i) Construct the angles of the following measurements: 30 o
Answer:
Steps to construction to follow:
Step 1: Draw a ray OY.
Step 2: Now, take A as a center and take any radius, then draw an arc AB cutting OY at A.
Step 3: Take A and B as centres, draw 2 arcs are marked intersecting each other at X and hence,
the bisector of is constructed. Aak
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Thus, is the required angle.
Q3.Construct the angles of the following measurements: (i) 30o
Edit Q
Q3 (ii) Construct the angles of the following measurements:
Answer:
Steps to construction to follow:
Step 1: Draw a ray OY.
Step 2: Now, take A as a centre and take any radius, then mark a point B on the arc Aakas
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Step 3: Take B as a centre with the same radius, mark a point C on the arc ABC.
Step 4: Now, taking B and C as centres simultaneously, then draw an arc from each centre
intersecting each other at a point X. Then join X and O and a ray making an angle with OY is
formed.
Let the arc AC touches OX at E
Step 5: Now, with A and E as centres, mark 2 arcs which intersect each other at D and we obtain
the bisector of the angle . Aak
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Q3 (iii) Construct the angles of the following measurements: 15 o
Answer:
Steps to construction to follow:
Step 1: Draw a ray OY.
Step 2: Now, take A as a centre and take any radius, draw an arc AB which cuts OY at A.
Step 3: Take A and B as a centre, then mark 2 arcs which intersect each other at X and Hence,
the bisector is constructed of .
Step 4: Now, with A and E as centres, mark 2 arcs which intersect each other at D and we obtain
the bisector of the angle .
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Thus, the angle of is obtained which is
Q4 (i) Construct the following angles and verify by measuring them by a protractor: 75 o
Answer:
Steps to construction to follow: Aakas
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Step 1: Draw a ray OY.
Step 2: Now, taking O as the centre draw an arc ABC.
Step 3: On taking A as a centre, draw two arcs B and C on the arc ABC.
Step 4: Now, taking B and C as centres, arcs are made to intersect at point E and
the is constructed.
Step 5: Taking A and C as centres, arcs are made to intersect at D.
Step 6: Now, join OD and hence, is constructed.
Hence, is the required angle.
Q4 (ii) Construct the following angles and verify by measuring them by a protractor: 105 o
Answer:
The steps of construction to be followed: Aak
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Step 1: Draw a ray OY.
Step 2: Then, taking O as a centre, draw an arc ABC.
Step 3: Now, with A as a centre, draw two arcs B and C which are made on the arc ABC.
Step 4: Taking B and C as centres simultaneously, arcs are made to intersect at E
and is constructed.
Step 5: With B and C as centres, arcs are made to intersect at X.
Step 6: Join the OX and we get is constructed.
Thus, the angle is
Q4 (iii) Construct the following angles and verify by measuring them by a protractor: 135 o
Answer:
The steps of construction to be followed: Aakas
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Step 1: Draw a ray DY.
Step 2: Draw an arc ACD with O as a center.
Step 3: Now, with A as a centre, draw two arcs B and C on the arc ACD.
Step 4: Taking B and C as centres, arcs are made to intersect at E and the angle formed
is .
Step 5: Take F and D as centres, draw arcs to intersect at point X or the bisector of angle EOD is
made.
Step 6: Join OX and the is made.
Hence, the angle required is .
Q5 Construct an equilateral triangle, given its side and justify the construction.
Answer:
The following steps to make an equilateral triangle: Aakas
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Step 1: Draw a line segment AB = 4 cm.
Step 2: With A and B as centres, make two arcs in the line segment AB. Mark it as D and E
respectively.
Step 3: Now, with D and E as centres, make the two arcs cutting the previous arcs respectively,
and forming an angle of each.
Step 4: Extend the lines of A and B until they intersect each other at point C. Aakas
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Hence, triangle constructed is ABC which is equilateral.
Now, Justification ;
Since the angles constructed are of each, so the third angle will also be .
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Excercise: 11.2
Q1 Construct a triangle ABC in which , and .
Answer:
The steps of construction are as follows:
Step 1: Draw a line segment BC of 7cm length. Taking the help of protractor make
an .
Step 2: Now, cut a line segment BD having 13 cm on BX which is
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Step 3: Now, join CD.
Step 4: Draw a perpendicular bisector of CD to intersect BD at a point A. Join AC. Then ABC is
the required triangle
Hence, the required triangle is ABC.
Q2 Construct a triangle ABC in which , ° and
Answer:
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The steps of construction to be followed:
Step 1: Draw a line segment BC = 8cm and make an angle of at point B i.e.,
Step 2: Now, cut the line segment BD = 3.5 cm on ray BX. i.e. .
Step 3: Join CD and draw a perpendicular bisector of CD i.e., PQ.
Step 4: Let the perpendicular bisector of CD intersects BX at point A. Then,
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Step 5: Join AC, to get the required triangle
Q3 Construct a triangle PQR in which , and .
Answer:
The steps of construction to be followed:
Step 1: Draw a ray QX and cut off a line segment QR which is equal to 6 cm in length.
Step 2: With an angle of with QR, construct a ray QY and extend it to form a line YQY'.
Step 3: Now, cut off a line segment QS equal to 2 cm from QY' and join RS.
Y
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Y'
Step 4: Draw a perpendicular bisector of RS which intersects QY at a point P.
Step 5: Join PR to get the required triangle
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Q4 Construct a triangle XYZ in which , °
and .
Answer:
The steps of construction to be followed:
Step 1: For given , a line segment is drawn.
Step 2: At points, P and Q angles of and are constructed
respectively.
Step 3: Now, bisects the angle RPQ and SQP. The bisectors of these angles intersect each other
at a point X.
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Step 4: Construct the perpendicular bisector of PX and QX, name them as TU and WV
respectively.
Step 5: Let the bisector TU intersect PQ at Y and bisector WV intersect PQ at Z. Then XY and
ZY are joined.
Therefore, is the required triangle.
Q5 Construct a right triangle whose base is 12cm and sum of its hypotenuse and other side is 18
cm.
Answer:
The steps of construction to follow:
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Step 1: Draw a ray BX and Cut off a line segment from the ray.
Step 2: Now, construct an angle .
Step 3: Cut off a line segment BD of length 18 cm on BY. Then join the CD.
Step 4: Now, construct a perpendicular bisector of CD which intersects BD at A and AC is
joined.
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Thus, the constructed triangle is ABC.
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