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# Short Answer Type Questions- Constructions Class 10 Notes | EduRev

## Class 10 : Short Answer Type Questions- Constructions Class 10 Notes | EduRev

The document Short Answer Type Questions- Constructions Class 10 Notes | EduRev is a part of the Class 10 Course Class 10 Mathematics by VP Classes.
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Q1. Draw a circle of diameter 6.4 cm. Then draw two tangents to the circle from a point P at a distance 6.4 cm from the centre of the circle.

Sol. Steps of construction:

I. Draw a circle with centre O and radius = 6.4/2 cm or 3.2 cm.
II. Mark a point P outside the circle such that OP = 6.4 cm.
III. Join OP.
IV. Bisect OP such that its mid point is at M.
V. With centre M and radius OM, draw a circle intersecting the given circle at A and B.
VI. Join PA and PB. Thus, PA and PB are the two tangents to the given circle.

Q2. Draw a circle of radius 3.4 cm. Draw two tangents to it inclined at an angle of 60Â° to each other:

Sol. Steps of construction:
I. Draw a circle with centre O and radius as 3.4 cm.
II. Draw two radii OA and OB such that âˆ AOB = 120Â°.
III. Draw perpendiculars at A and B such that these perpendiculars meet at P.
Obviously, âˆ APB = 60Â°. [Using Angle sum property of a quadrilateral]
IV. Thus, PA and PB are the required tangents to the given circle.

Q3. Draw Î” ABC in which AB = 3.8 cm, âˆ  B = 60Â° and median AD = 3.6 cm. Draw another triangle ABâ€™Câ€™ similar to the first such that AB = (4/3)AB.

Sol. Steps of construction:

I. Draw  AB = 3.8 cm.
II. Construct âˆ ABY = 60Â°.
III. With centre A and radius as 3.6 cm mark a ray to intersect BY at D.
IV. With centre D and radius BD, mark an arc to intersect BY at C.
V. Join CA. Thus, ABC is a triangle.
VI. Draw a ray AX, such that âˆ BAX is an acute angle.
VII. Mark 4 points X1, X2, X3 and X4 such that AX1 = X1X2 = X2 X3 = X3 X4.
VIII. Join X3B.
IX. Through X4 draw X4Bâ€™ y X3B X. Through Bâ€™ draw Bâ€™ Câ€™ â•‘ BC where Câ€² lies on AC (produced).
Thus, Î”Câ€™AB is the required triangle.

Q4. Draw an equilateral triangle of height 3.6 cm. Draw another triangle similar to it such that its side is 2/3 of the side of the first.

Sol. Steps at construction:
I. Draw a line segment RS.
II. Mark a point Y on it.
III. Through Y, draw YZ âŠ¥ RS
IV. Mark a point A on YZ such that YA = 3.6 cm
V. At A draw âˆ YAB = 30Â° such that the point B is on RS.
VI. With centre A and radius = AB, mark a point C on RS.
VII. Join AC.
VIII. Draw a ray BX such that âˆ CBX is an acute angle.
IX. Mark three points X1, X2, X3 such that AX1 = X1X2 = X2 X3.
X. Join X3 and C.
XI. Through X2 draw X2Câ€² â•‘ X3C.
XII. Through Câ€² draw Câ€² Aâ€² â•‘ CA.
Thus, âˆ†Aâ€² BCâ€² is the required triangle.

Q5. Draw an isosceles âˆ† ABC, in which AB = AC = 5.6 cm and âˆ  ABC = 60Â°. Draw another âˆ† ABâ€² Câ€² similar to âˆ†ABC such that ABâ€² = (2/3) AB.

Sol. Steps of Construction:
I. Draw a ray BD.
II. Through B, draw another ray BE such that âˆ DBE = 60Â°.
III. Cut off BA = 5.6 cm.
IV. With A as centre and radius 6 cm, mark an arc intersecting BD at C.
V. Join A and C to get Î”ABC.
VI. Draw a ray BX such that âˆ CBX is an acute angle.
VII. Mark three point X1, X2 and X3 such that BX1 = X1X2 = X2X3.
VIII. Join X3 and C.
IX. Through X2 draw X2Câ€² â•‘ X3C
X. Through Câ€² , draw Câ€² Aâ€² â•‘ CA
Thus, Î” Aâ€² BCâ€² is the required triangle.

Q6. Construct an isosceles triangle whose base is 9 cm and altitude is 5 cm. Then construct another triangle whose sides are 3/4 of the corresponding sides of the first isosceles triangle.

Sol. Steps of construction:
I. Construct a Î”ABC such that AB = AC, BC = 9 cm and altitude AD = 5 cm.
II. Through B, draw a ray BX such that âˆ CBX is an acute angle.
III. Mark 4 equal points X1, X2, X3 and X4 on BX. such that BX1 = X1 X2 = X2 X3 = X3 X4
IV. Join X4 and C.
V. Through X3, draw X3Câ€² â•‘ X4C, intersecting BC in Câ€².
VI. Through Câ€² , draw Câ€² Aâ€² â•‘ CA, intersecting AB in Aâ€².
Thus, Î”Aâ€²BCâ€² is the required triangle.

Q7. Draw a line segment AB of length 7 cm. Taking A as centre draw a circle of radius 3 cm and taking B as centre, draw another circle of radius 2.5 cm. Construct tangents to each circle from the centre of the other circle.

Sol. Steps of construction:
I. Draw a line segment AB = 7 cm
II. With centre A and radius 3 cm, draw a circle.
III. With centre B and radius 2.5 cm, draw another circle.
IV. Bisect AB and let M be the mid point of AB.
V. With centre M and radius AM, draw a circle intersecting the two circles in P,Q and R,S.
VI. Join AP, AQ, BR and BS.
Thus, AP, AQ, BR and BS are required tangents.

Q8. Construct a Î” ABC in which BC = 6.5 cm, AB = 4.5 cm and âˆ ABC = 60Â°. Construct a triangle similar to this triangle whose sides are 3/4 of the corresponding sides of the Î”ABC.

Sol. Steps of construction:

I. Construct the Î”ABC such that AB = 4.5 cm, âˆ B = 60Â° and BC = 6.5 cm.
II. Construct an acute angle âˆ BAX.
III. Mark 4 points X1, X2, X3 and X4 on AX such that AX1 = X1X2 = X2X3 =X3X4.
IV. Join X4 and B.
V. Draw X3 Bâ€² â•‘ BC, meeting AC at Câ€² .
Thus, Î”Câ€²ABâ€² is the required Î”.

Q9. Draw a right triangle in which sides (other than hypotenuse) are of lenghts 8 cm and 6 cm. Then construct another triangle whose sides are 3/4 times the corresponding sides of the first triangle.

Sol. Steps of construction:
I. Draw a Î”ABC such that AB = 8 cm, âˆ B = 90Â° and BC = 6 cm.
II. Construct an acute angle âˆ BAX.
III. Mark 4 points X1, X2, X3 and X4 on AX such that AX1 = X1X2, = X2X3 = X3X4.
IV. Join X4 and B.
V. Draw X3Bâ€² â•‘ X4B.
VI. Draw Bâ€² Câ€² â•‘ BC.
Thus, Î”ABâ€²Câ€² is the required rt Î”.

Q10. Construct a Î”ABC in which BC = 5 cm, CA = 6 cm and AB = 7 cm. Construct a Î”Aâ€² BCâ€² similar to Î”ABC, each of whose sides are 7/5 times the corresponding sides of Î”ABC.

Sol. Steps of construction:
I. Construct Î”ABC such that: BC = 5 cm, CA = 6 cm and AB = 7 cm.
II. Draw a ray BX such that âˆ CBX is an acute angle.
III. Mark 7 points X1, X2, .............. X7 such that: BX1 = X1X2, = X2X3 = X3X4 = X4X5 = X5X6 = X6X7
IV. Join X7 and C.
V. Draw a line through X5 parallel to X7C to meet BC extended at Câ€² .
VI. Through Câ€² , draw a line parallel to CA to meet BA extended at Aâ€² .

Thus, Î”Aâ€²BCâ€² is the required triangle.

Q11. Construct a triangle with sides 4 cm, 5 cm and 7 cm. Then construct a triangle similar to it whose sides are 2/3 of the corresponding sides of the given triangle.

Sol. Steps of construction:
I. Construct the Î”ABC such that BC = 7 cm, CA = 5 cm and BA = 4 cm.
II. Draw a ray BX such that âˆ CBX is an acute angle.
III. Mark three points X1, X2 and X3 on BX such that: BX1 = X1 X2 = X2 X3
IV. Join X3 and C.
V. Draw X2Câ€² â•‘ X3C.
VI. Draw Câ€² Aâ€² â•‘ CA

Thus, Î”Aâ€² BCâ€² is the required triangle.

Q12. Construct a Î”ABC in which AB = 6.5 cm, âˆ  B = 60Â° and BC = 5.5 cm. Also construct a triangle ABâ€²Câ€² similar to Î”ABC whose each side is 3/2 times the corresponding side of the Î”ABC.

Sol. Steps of construction:
I. Construct a Î”ABC such that AB = 6.5 cm, âˆ B = 60Â° and BC = 5.5 cm.
II. Draw a ray AX making an acute angle âˆ BAX.
III. Mark three points X1, X2, X3 on the ray AX such that AX1 = X1 X2 = X2 X3
IV. Join X2 and B.
V. Draw X3Bâ€² â•‘ X2B such that Bâ€² is a point on extended AB.
VI. Join Bâ€² Câ€² y BC such that Câ€² is a point on AC (extended).
Thus, Î”Câ€² ABâ€² is the required triangle.

Q13. Draw a Î”ABC with side BC = 6 cm, AB = 5 cm and âˆ  ABC = 60Â°. Construct Î”ABâ€² Câ€² similar to Î”ABC such that sides of Î”ABâ€²Câ€² are 3/4 of the corresponding sides of Î”ABC.

Sol. Steps of construction:

I. Construct the given Î”ABC.
II. Draw a ray AX such that âˆ BAC is an acute angle.
III. Mark 4 points X1, X2, X3 and X4 on   such that AX1 = X1 X2 = X2 X3 = X3 X4.
IV. Join X4 B.
V. Draw X3Bâ€² â•‘ X4B
VI. Through Bâ€² draw Bâ€² Câ€² â•‘ BC.
Thus, Î”Bâ€² ACâ€² is the required triangle.

Q14. Draw a circle of radius 3 cm. From a point P, 6 cm away from its centre, construct a pair of tangents to the circle. Measure the lengths of the tangents.

Sol. Steps of construction:

I. Draw the given circle such that its centre is at O and radius = 3 cm.
II. Mark a point P such that OP = 6 cm.
III. Bisect OP. Let M be the mid point of OP.
IV. Taking M as centre and OM as radius draw a circle intersecting the given circle at A and B.
V. Join PA and PB.
Thus, PA and PB are the required tangents to the given circle.

Q15. Construct a triangle whose perimeter is 13.5 cm and the ratio of the three sides is 2 : 3 : 4.

Sol. Steps of construction:
I. Draw a line PQ = 13.5 cm
II. At P, draw a ray PR making a convenient acute angle â€“QPR with PQ.
III. On PR mark (2 + 3 + 4), 9 points at equal distances.
IV. Join Q and the mark 9.
V. Through the points 2 and 5 draw lines 2-A and 5-B parallel to 9-Q. Let these lines meet PQ at A and B respectively.
VI. With A as centre and radius = AP, draw an arc.
VII. With B as centre and radius = BQ, draw another arc which intersects the arc of step
VI at C.
VIII. Join CA and CB.
ABC is the required triangle.

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