Constructions for Class 10

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Construction refers to the process of creating geometric figures using specific tools such as a compass, straightedge, protractor or ruler, according to precise instructions or rules. The goal of construction is to accurately draw or create geometric shapes, angles or lines, often following a set of mathematical principles or constructions. This chapter includes the construction of a tangent to a circle through a point on its circumference, the construction of the pair of tangents from an external point to a circle, the construction of circumscribed and inscribed circles of a triangle and the construction of circumscribing and inscribing a circle on a regular hexagon.

  • Constructing a tangent to a circle from a point on its circumference
  • Constructing tangents to a circle from a point outside the circle
  • Constructing a circumscribing circle of a triangle
  • Constructing an inscribed circle of a triangle
  • Constructing a circumscribing circle of a regular hexagon
  • Constructing an inscribing circle of a regular hexagon
  • Construction 1: Constructing a tangent to a circle from a point on its circumference

    Step 1: Draw a circle using point O as the centre, with a radius measuring r cm.

    Step 2: Take a point P on it.

    Step 3: Join OP.

    Step 4: Construct a line APB that forms a right angle with the line OP.

    APB is the required tangent.

    cmo-constructions-c10-1

    Construction 2: Constructing tangents to a circle from a point outside the circle.

    Let O be the centre of the given circle and P be a point outside the circle.

    Step 1: Join O and P.

    Step 2: Draw a circle using OP as the diameter. It intersects the given circle at points A and B.

    Step 3: Join PA and PB.

    PA and PB are the required tangents.

    Note:

    → The angle formed between the radius and the tangent where they meet is 90°.

    → Tangents from an exterior point are always equal, that is PA =PB.

    → In △APO, ∠PAO = 90°. Using Pythagoras theorem,

    (PA)2 + (OA)2 = (OP)2

    (PA)2 = (OP)2 − (OA)2

    PA = √(OP)2 − (OA)2

    cmo-constructions-c10-2

    Construction 3: Constructing a circumscribing circle of a triangle.

    Consider the triangle ABC as given.

    Step 1: Draw the perpendicular bisectors of any two sides of a triangle intersecting each other at O.

    Let the perpendicular bisectors of AB and AC be drawn.

    Step 2: Taking O as the centre and OA as the radius, draw a circle.

    The circle drawn is the required circle.

    Note:

    → O is the circumcentre and OA, OB and OC are the circumradius.

    cmo-constructions-c10-3

    Construction 4: Constructing an inscribed circle of a triangle.

    Consider the triangle ABC as given.

    Step 1: Draw the angle bisectors of any two angles of a triangle meeting at I.

    Let the bisectors of angles A and B be drawn.

    Step 2: Draw a perpendicular to any side of the triangle from point I.

    Let ID be the perpendicular drawn.

    Step 3: Taking I as the centre and ID as the radius, draw a circle.

    The circle drawn is the required circle.

    Note:

    → I is the incentre and IA, IB and IC are the in radius.

    cmo-constructions-c10-4

    Construction 5: Constructing a circumscribing circle of a regular hexagon.

    Step 1: Draw the regular hexagon ABCDEF with the given data.

    Step 2: Draw the perpendicular bisectors of any two sides of the regular hexagon intersecting each other at O.

    Let the perpendicular bisectors of AB and AF be drawn.

    Step 3: Taking O as the centre and OA as the radius, draw a circle.

    The circle drawn is the required circle.

    Alternative Method

    Step 1: Draw the regular hexagon ABCDEF with the given data.

    Step 2: Draw any two main diagonals of the regular hexagon intersecting each other at O.

    Let the main diagonals AD and CF be drawn.

    Step 3: Taking O as the centre and OF as the radius, draw a circle.

    The circle drawn is the required circle.

    cmo-constructions-c10-5
    cmo-constructions-c10-6

    Construction 6: Constructing an inscribing circle of a regular hexagon.

    Step 1: Draw the regular hexagon ABCDEF with the given data.

    Step 2: Draw the angle bisectors of any two angles of the regular hexagon intersecting each other at I.

    Let the angle bisectors of angles A and B be drawn.

    Step 3: Draw IP perpendicular to AB from point I.

    Step 4: Taking I as the centre and IP as the radius, draw a circle.

    The circle drawn is the required circle.

    cmo-constructions-c10-7

    Note:

    → The interior angle of the regular polygon of n sides is given by:

    Each interior angle = 2n − 4n n  × 90°

    Example 1: If two tangents are drawn from an external point 7.5 cm from the centre of the circle of diameter 9 cm, then what is the length of each tangent?

    a) 6.1 cm
    b) 6.5 cm
    c) 6 cm
    d) 6.2 cm

    Answer: c) 6 cm

    Explanation: We know that the radius of the given circle is half the diameter.

    Thus, Radius = 9/2 = 4.5 cm

    Construct the tangents using the following steps of construction:

    Step 1: Draw a circle with a centre O and a radius of 4.5 cm.

    Step 2: From point O, take another point P and the distance between O and P is 7.5 cm.

    Step 3: Draw a bisector of OP that cuts OP at M.

    Step 4: Take M as the centre and OM as the radius and draw another circle intersecting the given circle at A and B.

    Step 5: Join AP and BP.

    AP and BP are the required tangents.

    cmo-constructions-c10-8

    In △AOP, ∠PAO = 90°. Using Pythagoras theorem,

    (PA)2 + (OA)2 = (OP)2

    (PA)2 = (OP)2 − (OA)2

    PA = √[(OP)2 − (OA)2]

    We know that OP = 7.5 cm, OA = 4.5 cm

    → PA = √[(7.5)2 − (4.5)2]

    → PA = √(56.25 − 20.25)

    → PA = √36

    → PA = 6 cm

    Since tangents from an exterior point are always equal, that is PA = PB.

    → PB = 6 cm

    Example 2: Arrange the given steps for constructing a circle inscribing an equilateral triangle with side 5 cm in CORRECT order.

    Steps of Construction:

    Step 1: With B and C as centre and radius of 5 cm, draw two arcs intersecting each other at point A.
    Step 2: From O, draw OL perpendicular to BC.
    Step 3: With O as the centre and OL as the radius, draw a circle touching the triangle's sides.
    Step 4: Join AB and AC.
    Step 5: Draw a line segment BC of length 5 cm.
    Step 6: Draw angle bisectors of B and C intersecting each other at O. 

    a) 5 - 4 - 1 - 6 - 3 - 2
    b) 5 - 1 - 6 - 4 - 2 - 3
    c) 5 - 1 - 4 - 6 - 3 - 2
    d) 5 - 1 - 4 - 6 - 2 - 3

    Answer: d) 5 - 1 - 4 - 6 - 2 - 3

    Explanation: Construct the figure using the given steps:

    cmo-constructions-c10-9

    Steps of Construction:

    Step 1: Draw a line segment BC of length 5 cm.
    Step 2: With B and C as centre and radius of 5 cm, draw two arcs intersecting each other at point A.
    Step 3: Join AB and AC.
    Step 4: Draw angle bisectors of B and C intersecting each other at O. 
    Step 5: From O, draw OL perpendicular to BC.
    Step 6: With O as the centre and OL as the radius, draw a circle touching the triangle's sides.

    Thus, this is the required incircle.

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