Study MaterialsImportant QuestionsImportant Questions for Class 10 Maths Chapter 11 Constructions

Important Questions for Class 10 Maths Chapter 11 Constructions

Important Questions for Class 10 Maths Chapter 11 Constructions

Constructions Class 10 Important Questions Short Answer-I (2 Marks)

Question 1.
Draw a line segment of length 6 cm. Using compasses and ruler, find a point P on it which divides it in the ratio 3 : 4. (2011D)
Solution:
Important Questions for Class 10 Maths Chapter 11 Constructions 1
Hence, PA : PB = 3 : 4

Question 2.
Draw a line segment AB of length 7 cm. Using ruler and compasses, find a point P on AB such that \(\frac{\mathbf{A} \mathbf{P}}{\mathbf{A B}}=\frac{3}{5}\). (2011OD)
Solution:
AB = 7 cm, AB = \(\frac{\mathbf{A} \mathbf{P}}{\mathbf{A B}}=\frac{3}{5}\) … [Given
∴ AP : PB = 3 : 2
Important Questions for Class 10 Maths Chapter 11 Constructions 2
Hence, AP : AB = 3 : 5 or \(\frac{\mathbf{A} \mathbf{P}}{\mathbf{A B}}=\frac{3}{5}\)

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    Constructions Class 10 Important Questions Short Answer-II (3 Marks)

    Question 3.
    Draw a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60°. Then construct a triangle whose sides are \(\frac{5}{7}\) times the corresponding sides of ∆ABC. (2011D)
    Solution:
    In ∆ABC
    AB = 5 cm
    BC = 6 cm
    ∠ABC = 60°
    Important Questions for Class 10 Maths Chapter 11 Constructions 3
    Hence, ∆A’BC’ is the required ∆.

    Question 4.
    Construct a triangle with sides 5 cm, 5.5 cm and 6.5 cm. Now construct another triangle, whose sides are \(\frac{3}{5}\) times the corresponding sides of the given triangle. (20140D)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 4
    ∴ ∆AB’C’ is the required ∆.

    Question 5.
    Construct a right triangle in which the sides, (other than the hypotenuse) are of length 6 cm and 8 cm. Then construct another triangle, whose sides are \(\frac{3}{5}\) times the corresponding sides of the given triangle. (2012D)
    Solution:
    Here AB = 8 cm, BC = 6 cm and
    Ratio = \(\frac{3}{5}\) of corresponding sides
    Important Questions for Class 10 Maths Chapter 11 Constructions 5
    ∴ ∆AB’C’ is the required triangle.

    Question 6.
    Draw a triangle PQR such that PQ = 5 cm, ∠P = 120° and PR = 6 cm. Construct another triangle whose sides are \(\frac{3}{4}\) times the corresponding sides of ∆PQR. (2011D)
    Solution:
    In ∆PQR,
    PQ = 5 cm, PR = 6 cm, ∠P = 120°
    Important Questions for Class 10 Maths Chapter 11 Constructions 6
    ∴ ∆POʻR’ is the required ∆.

    Question 7.
    Draw a triangle ABC with BC = 7 cm, ∠B = 45° and ∠C = 60°. Then construct another triangle, whose sides are \(\frac{3}{5}\) times the corresponding sides of ∆ABC. (2012OD)
    Solution:
    Here, BC = 7 cm, ∠B = 45°, ∠C = 60° and ratio is \(\frac{3}{5}\) times of corresponding sides
    Important Questions for Class 10 Maths Chapter 11 Constructions 7
    ∴ ∆A’BC’ is the required triangle.

    Question 8.
    Construct a triangle with sides 5 cm, 4 cm and 6 cm. Then construct another triangle whose sides are \(\frac{2}{3}\) times the corresponding sides of first triangle. (2013D)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 8
    Steps of Construction:

    • Draw ∆ABC with AC = 6 cm, AB = 5 cm, BC = 4 cm.
    • Draw ray AX making an acute angle with AÇ.
    • Locate 3 equal points A1, A2, A3 on AX.
    • Join CA3.
    • Join A2C’ || CA3.
    • From point C’ draw B’C’ || BC.

    ∴ ∆AB’C’ is the required triangle.

    Question 9.
    Draw a pair of tangents to a circle of radius 3 cm, which are inclined to each other at an angle of 60°. (2011OD)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 9
    ∴ PA & PB are the required tangents.

    Question 10.
    Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm. (2013D)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 10
    Steps of Construction:
    Draw two circles with radius OA = 4 cm and OP = 6 cm with O as centre. Draw ⊥ bisector of OP at M. Taking M as centre and OM as radius draw another circle intersecting the smaller circle at A and B and touching the bigger circle at P. Join PA and PB. PA and PB are the required tangents.
    Verification:
    In rt. ∆OAP,
    OA2 + AP2 = OP2 … [Pythagoras’ theorem
    (4)2 + (AP)2 = (6)2
    AP2 = 36 – 16 = 20
    AP = + \(\sqrt{20}=\sqrt{4 \times 5}\)
    = \(2 \sqrt{5}\) = 2(2.236) = 4.472 = 4.5 cm
    By measurement, ∴ PA = PB = 4.5 cm

    Question 11.
    Draw a pair of tangents to a circle of radius 4.5 cm, which are inclined to each other at an angle of 45°. (2013OD)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 11
    Draw ∠AOB = 135°, ∠OAP = 90°, ∠OBP = 90°
    ∴ PA and PB are the required tangents.

    Question 12.
    Draw two tangents to a circle of radius 3.5 cm, from a point P at a distance of 6.2 cm from its cehtre. (2013OD)
    Solution:
    OP = OC + CP = 3.5 + 2.7 = 6.2 cm
    Important Questions for Class 10 Maths Chapter 11 Constructions 12
    Hence AP & PB are the required tangents.

    Question 13.
    Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle. (2014D, 2015OD)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 13
    Steps of Construction:

    • Draw BC = 8 cm.
    • From B draw an angle of 90°.
    • Draw an arc BA = 6 cm cutting the angle at A.
    • Join AC. ∴ ∆ABC is the required ∆.
    • Draw ⊥ bisector of BC cutting BC at M.
    • Take Mas centre and BM as radius, draw a circle.
    • Take A as centre and AB as radius draw an arc cutting the circle at E. Join AE.

    AB and AE are the required tangents.
    Justification: ∠ABC = 90° …[Given
    Since, OB is a radius of the circle.
    ∴ AB is a tangent to the circle.
    Also, AE is a tangent to the circle.

    Question 14.
    Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle. (2014OD)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 14
    Draw two circles on A and B as asked.
    Z is the mid-point of AB.
    From Z, draw a circle taking ZA = ZB as radius,
    so that the circle intersects the bigger circle at M and N and smaller circle at X and Y.
    Join AX and AY, BM and BN.
    BM, BN are the reqd. tangents from external point B.
    AX, AY are the reqd. tangents from external point A.
    Justification:
    ∠AMB = 90° …[Angle in a semi-circle
    Since, AM is a radius of the given circle.
    ∴ BM is a tangent to the circle
    Similarly, BN, AX and AY are also tangents.

    Constructions Class 10 Important Questions Long Answer (4 Marks).

    Question 15.
    Draw a triangle ABC with side BC = 7 cm, ∠B = 45° and ∆A = 105°. Then construct a triangle whose sides are \(\frac{3}{5}\)times the corresponding sides of ∆ABC. (2011D)
    Solution:
    In ∆ABC, ∠A + ∠B + ∠C = 180° … [angle sum property of a ∆
    105° + 45° + C = 180°
    ∠C = 180° – 105° – 45o = 30°
    BC = 7 cm
    Important Questions for Class 10 Maths Chapter 11 Constructions 15
    ∴ ∆A’BC’ is the required ∆.

    Question 16.
    Draw a triangle ABC with side BC = 6 cm, ∠C = 30° and ∠A = 105°. Then construct another triangle whose sides are \(\frac{2}{3}\) times the corresponding sides of ∆ABC. (2012D)
    Solution:
    Here, BC = 6 cm, ∠A = 105o and ∠C = 30°
    Important Questions for Class 10 Maths Chapter 11 Constructions 16
    In ∆ABC,
    ∠A + ∠B + ∠C = 180° …[Angle-sum-property of a ∆
    105° + ∠B + 30o = 180°
    ∠B = 180° – 105° – 30o = 45°
    ∴ ∆A’BC’ is the required ∆.

    Question 17.
    Draw a triangle with sides 5 cm, 6 cm and 7 cm. Then construct another triangle whose sides are \(\frac{2}{3}\) times the corresponding sides of the first triangle. (2012OD)
    Solution:
    Here, AB = 5 cm, BC = 7 cm, AC = 6 cm and ratio is \(\frac{2}{3}\) times of corresponding sides.
    Important Questions for Class 10 Maths Chapter 11 Constructions 17
    ∴ ∆A’BC’ is the required triangle.

    Question 18.
    Construct an isosceles triangle whose base is 6 cm and altitude 4 cm. Then construct another triangle whose sides are \(\frac{3}{4}\) times the corresponding sides of the isosceles triangle. (2015D)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 18
    ∴ ∆A’BC’ is the required triangle.

    Question 19.
    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 cm. Construct tangents to each circle from the centre of the other circle. (2015D)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 19
    Draw two circles on A and B as asked.
    Z is the mid-point of AB.
    From Z, draw a circle taking ZA = ZB as radius,
    so that the circle intersects the bigger circle at M and N and smaller circle at X and Y.
    Join AX and AY, BM and BN.
    BM, BN are the required tangents from external point B.
    AX, AY are the required tangents from external point A.
    Justification:
    ∠AMB = 90° …[Angle in a semi-circle
    Since, AM is a radius of the given circle.
    ∴BM is a tangent to the circle
    Similarly, BN, AX and AY are also tangents.

    Question 20.
    Construct a ∆ABC in which AB = 6 cm, ∠A = 30° and ∆B = 60°. Construct another ∆AB’C’ similar to ∆ABC with base AB’ = 8 cm. (2015OD)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 20
    Steps of construction:

    • Draw a ∆ABC with side AB = 6 cm, ∠A = 30° and ∠B = 60°.
    • Draw a ray AX making an acute angle with AB on the opposite side of point C.
    • Locate points A1, A2, A3 and A4 on AX.
    • Join A3B. Draw a line through A4 parallel to A3B intersecting extended AB at B’.
    • Draw a line parallel to BC intersecting ray AY at C’.

    Hence, ∆AB’C’ is the required triangle.

    Question 21.
    Construct a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60°. Now construct another triangle whose sides are \(\frac{5}{7}\) times the corresponding sides of ∆ABC. (2015OD)
    Solution:
    In ∆ABC, AB = 5 cm; BC = 6 cm; ∠ABC = 60°
    Important Questions for Class 10 Maths Chapter 11 Constructions 21
    ∴ ∆A’BC’ is the required ∆.

    Question 22.
    Construct a triangle ABC in which BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct another triangle whose sides are \(\frac{3}{4}\) times the corresponding sides of ∆ABC. (2016D)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 22
    Steps of Construction:

    • Draw ∆ABC with the given data.
    • Draw a ray BX downwards making an acute angle with BC.
    • Locate 4 points B1, B2, B3, B4, on BX, such that BB1 = B1B2 = B2B3 = B3B4.
    • Join CB4.
    • From B3 draw a line C’B3 || CB4 intersecting BC at C’.
    • From C’ draw A’C’ || AC intersecting AB at B’.

    Then ∆AB’C’ in the required triangle.
    Justification:
    Important Questions for Class 10 Maths Chapter 11 Constructions 23

    Question 23.
    Draw a triangle ABC with BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a triangle whose sides are \(\frac{4}{5}\) times the corresponding sides of ∆ABC. (2016D)
    Solution:
    In ∆ABC, ∠A + ∠B + ∠C = 180° ..[Angle-sum-property of a ∆
    105° + 45° + ∠C = 180°
    ∠C = 180° – 105° – 45o = 30°
    Important Questions for Class 10 Maths Chapter 11 Constructions 24
    ∴ A’BC’ is the required triangle.

    Question 24.
    Draw an isosceles ∆ABC in which BC = 5.5 cm and altitude AL = 3 cm. Then construct another triangle whose sides are \(\frac{3}{4}\) of the corresponding sides of ∆ABC. (2016OD)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 29
    ∴ ∆ABC’ is the required ∆.

    Question 25.
    Draw a triangle with sides 5 cm, 6 cm and 7 cm. Then draw another triangle whose sides are \(\frac{4}{5}\) of the corresponding sides of first triangle. (2016OD)
    Solution:
    Important Questions for Class 10 Maths Chapter 11 Constructions 25
    ∴ ∆A’BC’ is the required ∆.

    Question 26.
    Draw two concentric cirlces of radii 3 cm and 5 cm. Construct a tangent to smaller circle from a point on the larger circle. Also measure its length. (2016D)
    Solution:
    We have, OD = 3 cm and OP = 5 cm
    Important Questions for Class 10 Maths Chapter 11 Constructions 27
    PA and PB are the required tangents
    By measurement PA = PB = 4 cm.

    Question 27.
    Draw a circle of radius 4 cm. Draw two tangents to the circle inclined at an angle of 60° to each other. (2013, 2016OD)
    Solution:
    Draw a circle with O as centre and radius 4 cm.
    Draw any ∠AOB = 120°. From A and B draw ∠PAO = ∠PBO = 90° which meet at P.
    Important Questions for Class 10 Maths Chapter 11 Constructions 28
    ∴ PA and PB are the required tangents.

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