Chapter: Constructions - Class 10

Constructions - Sub Topics

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

Solved Questions based on Construction

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.

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

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.

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.

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.

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.

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.

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:

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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Maths Related Topics for Class 10

Number System and Polynomials

Arithmetic Progressions

Quadratic and Linear Equations

Trigonometry

Coordinate Geometry

Triangles and its Properties

Circles and Area Related to Circles

Surface Areas and Volumes

Statistics

Probability

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