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.
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.
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}
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.
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.
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.
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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