ICSE Class 10 Maths Chapter 22 Constructions

Chapter 22

Constructions

Points To Remember

1. Length of Direct Common Tangents:

\[L = \sqrt{d^2 - (r_1 - r_2)^2}\]

where d is the difference between the centres of two circles and r₁, r₂ are the radii of the circles.

2. Length of Inverse Common Tangents:

\[L = \sqrt{d^2 - (r_1 + r_2)^2}\]

where d is the difference between the centres of two circles and r₁, r₂ are the radii of the circles.

3. Angle at the Centre of a Regular Polygon:

\[\theta = \frac{360°}{n}\]

where n is the number of sides of the regular polygon.

4. Circumcircle of a Triangle:

The circle passing through the vertices of a triangle is called the circumcircle of the triangle.

5. Circumcentre of a Triangle:

The point of intersection of the right bisectors of any two sides of a triangle is called its circumcentre.

6. Incircle of a Triangle:

A circle inside a triangle touching its sides, is called the incircle of a triangle.

7. Incentre of a Triangle:

The point of intersection of the bisectors of any two angles of a triangle is called the incentre of the triangle.

Teacher's Note

Understanding geometric constructions helps students develop spatial reasoning skills that are useful in architecture, engineering, and design fields.

Exercise 22 (A)

Q.1. Draw a circle of radius 3 cm. Take a point P on it. Using ruler and compasses only construct a tangent to the circle at the point P.

Sol. Steps of Construction:

(i) Draw a circle with O as centre and radius 3 cm.

(ii) Take a point P on it.

(iii) Join OP.

(iv) At P draw a perpendicular PT on OP and produce it both sides.

Then, TPT' is the required tangent to the circle.

Teacher's Note

Tangent construction is fundamental in understanding how curves behave and is applied in road design where tangent lines represent straight sections connecting curved highways.

Q.2. Draw a circle of radius 3-4 cm. Take a point P on it. Without using the centre of the circle, construct a tangent to the circle at the point P.

Sol. Steps of Construction:

(i) Draw a circle with radius 3-4 cm.

(ii) Take a point P on it.

(iii) Take two more points Q and R on it.

(iv) Join QP and PR and RQ.

(v) Draw angle QPT = angle PRQ.

(vi) Produce TP to T'.

Then T'PT is the required tangent to the circle.

Teacher's Note

This alternative construction method demonstrates that geometric properties can be discovered through angle relationships, similar to how surveyors find tangent lines without knowing the center point.

Q.3. Draw a circle of radius 2-7 cm. Mark its centre as O. Take a point P at distance of 5-3 cm from O. From the point P, draw two tangents to the circle. Measure the length of each.

Sol. Steps of Construction:

(i) Take a line segment OP = 5-3 cm.

(ii) At O, with a radius of 2-7 cm, draw a circle.

(iii) Bisect OP at M.

(iv) With centre M and OP as diameter, draw a circle intersecting the given circle at T and S.

(v) Join PT and PS. PT and PS are the required tangents to the circle.

On measuring them, PT = PS = 4-7 cm.

Teacher's Note

The property that tangents from an external point are equal is used in security systems where equal distances ensure balanced design.

Q.4. Draw a circle of radius 4 cm. Mark its centre as C and mark a point D such that CD = 7 cm. Using ruler and compasses only but not using the centre of the circle, construct two tangents from D.

Sol. Steps of Construction:

(i) Draw a circle of radius 4 cm with centre C.

(ii) Take a point D such that CD = 7 cm.

(iii) Through D, draw a secant AB intersecting the circle at A and B.

(iv) With DB as diameter and M, the mid-point of DB as centre, draw a semi-circle.

(v) From A, draw a perpendicular meeting the semi-circle at P.

(vi) With D as centre and DP as radius draw an arc intersecting the circle at T and S.

(vii) Join DT and DS.

DT and DS are the required tangents to the given circle.

Teacher's Note

This advanced construction technique shows how complex geometric relationships can be built from simpler elements, similar to how engineering problems are solved by breaking them into manageable parts.

Q.5. Draw a circle of radius 3-2 cm. Draw two tangents to it inclined at an angle of 60° with each other.

Sol. Steps of Construction:

(i) Draw a circle with centre O and radius 3-2 cm.

(ii) Draw a radius OS.

(iii) At O, draw angle of (180° - 60°) = 120° on OT.

(iv) At S and T, draw perpendiculars meeting each other at P.

Then, PT and PS are the required tangents inclined at an angle of 60°.

Teacher's Note

Tangent angles are critical in optical design where light rays must be directed at specific angles for lens and mirror systems.

Q.6. Draw a circle of radius 2-5 cm. Draw two tangents to it inclined at an angle of 45° to each other.

Sol. Steps of Construction:

(i) Draw a circle with centre O and radius 2-5 cm.

(ii) Draw a radius OS.

(iii) With OS and at O, draw another radius so that angle TOS = (180° - 45°) 135° with each other.

(iv) At S and S, draw perpendiculars meeting each other at P.

(v) TP and SP are the required tangents inclined at an angle of 45°.

Exercise 22 (B)

Q.1. Using ruler and compasses only, draw an equilateral triangle of side 4-5 cm and draw its circumscribed circle. Measure the radius of the circle.

Sol. Steps of Construction:

(i) Draw a line segment BC = 4-5 cm.

(ii) With centres B and C, draw arcs of radius 4-5 cm each intersecting each other at A.

(iii) Join AB and AC.

Triangle ABC is an equilateral triangle.

(iv) Draw the bisectors of side BC and AC intersecting each other at O.

(v) Join OB.

(vi) With centre O and radius OB, draw a circle which passes through A, B and C.

This is the required circle and on measuring its radius, it is 2-6 cm.

Teacher's Note

Circumscribed circles are used in manufacturing precision parts and designing regular structures like polygon-based architectural elements.

Q.2. Using ruler and compasses only, draw an equilateral triangle of height 6 cm and draw its inscribed circle. Measure the radius of the circle.

Sol. Steps of Construction:

(i) Draw a line XY and take a point D on it.

(ii) At D, draw a perpendicular and cut off DA = 6 cm.

(iii) At A, draw rays making an angle of 30° with AD both sides meeting the line XY at B and C respectively.

(iv) Draw the angle bisectors of B and C intersecting each other at I and from I, draw a perpendicular I D on BC.

(v) With I as centre and I D as radius, draw a circle which to makes the sides of the triangle internally. This is the required incircle and its radius, on measuring it is 2 cm.

Teacher's Note

Inscribed circles have practical applications in maximizing space usage, such as fitting circular storage tanks inside triangular floor plans.

Q.3. (i) Construct a triangle with sides 5 cm, 4 cm and 3 cm. Draw its circumcircle and measure its radius.

Q.3. (ii) Using a ruler and a pair of compasses only, construct:

(a) a triangle ABC, given AB = 4 cm, BC = 6 cm and angle ABC = 90°.

(b) a circle which passes through the points A, B and C and mark its centre as O.

Sol. (i) Steps of Construction:

(a) Draw a line segment BC = 5 cm.

(b) With centre B and radius 4 cm, and with centre C and radius 3 cm, draw arcs intersecting each other at A.

(c) Join AB and AC.

(d) Draw the perpendicular bisectors of sides AC and BC intersecting each other at O.

(e) We see that O lies on the side BC.

(f) With centre O and radius OB, draw a circle which will pass through A, B and C respectively.

On measuring the radius of the circle, it is 2-5 cm.

(ii) In triangle ABC, AB = 4 cm, BC = 6 cm, angle ABC = 90°

Steps of construction:

(a) Draw a line segment AB = 4 cm

(b) At B, draw a ray B x making an angle of 90° and cut off BC = 6 cm.

(c) Join AC.

(d) Draw the perpendicular bisectors of sides AB and AC which intersect each other at O.

(e) With centre O and radius equal to OB or OA or OC draw a circle which passes through A, B and C.

This is the required circle and on measuring the radius of the circle, it is 2-5 cm.

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