Mathematics Class 10 Unit 13: Practical Geometry – Circles Notes

This final unit moves from proving theorems to actually constructing circles and their related figures with a compass and straightedge. It covers locating the centre of a given circle, drawing a circle through three non-collinear points or through part of a given arc, and building the special circles attached to a triangle — the circumcircle (through all three vertices), the incircle (touching all three sides internally), and the escribed circle (touching one side and the other two sides extended).

The unit then extends to circumscribing and inscribing regular polygons — equilateral triangles, squares, and regular hexagons — about and within a given circle, and finally to a large family of tangent constructions: tangents to an arc without knowing its centre, tangents from a point on or outside a circle, tangents meeting at a given angle, and direct (external) and transverse (internal) common tangents to two circles of equal or unequal radii, including circles that already touch or intersect. Every construction in this unit rests on two repeated ideas from earlier units: the perpendicular bisector of a chord passes through the centre, and a tangent is always perpendicular to the radius at its point of contact.

Learning Objectives

• Locate the centre of a given circle and construct a circle through three non-collinear points
• Complete a circle when only part of its circumference is given, both by finding the centre and without finding it
• Circumscribe and inscribe a circle about and in a given triangle, and escribe a circle to a triangle
• Circumscribe and inscribe an equilateral triangle, a square, and a regular hexagon with respect to a given circle
• Construct a tangent to an arc or circle from a point on the circumference, outside it, or on the arc itself, without necessarily knowing the centre
• Construct direct (external) and transverse (internal) common tangents to two circles of equal or unequal radii, including circles that touch or intersect
• Construct a circle touching both arms of an angle, or touching two converging lines and passing through a given point between them

Key Concepts

13.1 Locating a Centre and Constructing Circles Through Points or Arcs

To locate the centre of a given circle, draw any two chords and construct the perpendicular bisector of each; since the perpendicular bisector of any chord always passes through the centre (Unit 9), the two bisectors intersect exactly at the centre O. The same idea, run in reverse, constructs a circle through three given non-collinear points A, B, and C: the perpendicular (right) bisectors of AB and BC meet at a single point O that is equidistant from all three points, and a circle centred at O with that radius passes through all three.

To complete a circle when only part of the circumference (an arc) is given, by finding the centre: mark four points C, D, E, F on the arc, draw chords CD and EF, and construct their perpendicular bisectors — these meet at the centre O, equidistant from every marked point, and the full circle can then be drawn with radius OA. A second method completes the circle WITHOUT finding the centre, by repeatedly constructing equal external angles between successive equal-length chords along the arc — each new equal angle and equal chord extends the circle's circumference a little further until it closes up.

13.2 Circles Attached to Triangles and Regular Polygons

The circumcircle of a triangle ABC passes through all three vertices; it is constructed by drawing the perpendicular bisectors of any two sides (say AB and AC) and taking their intersection O as the circumcentre, with circumradius OA = OB = OC. The incircle touches all three sides internally; it is constructed by bisecting two of the triangle's interior angles (say at B and C) to find the incentre O, then dropping a perpendicular OP to any side to get the in-radius OP. The escribed circle (e-circle) touches one side of the triangle externally and the other two sides extended; it is constructed by bisecting two EXTERIOR angles of the triangle to find the e-centre, then dropping a perpendicular to the opposite side for the e-radius.

An equilateral triangle can be circumscribed about a given circle (tangents drawn at three points spaced 120 degrees apart around the circle) or inscribed within it (joining three points on the circle that are each 120 degrees apart, found using arcs of the circle's own radius). A square is circumscribed by drawing two perpendicular diameters and tangents at all four endpoints, or inscribed by simply joining the four endpoints of two perpendicular diameters directly. A regular hexagon is circumscribed or inscribed similarly, using six points spaced 60 degrees apart around the circle — found by repeatedly stepping off arcs equal to the circle's own radius, since a chord equal to the radius always subtends a 60-degree central angle.

13.3 Tangent Constructions

A tangent can be drawn to an arc without knowing its centre in three cases: when the given point P is the arc's midpoint (construct the perpendicular bisector of the chord through the arc, then erect a right angle at P to that bisector); when P is an endpoint of the arc (use an auxiliary point and an equal-angle construction based on the tangent-chord angle being equal to the inscribed angle in the alternate segment); and when P is outside the arc entirely (using a semicircle construction on segment AP, similar to the construction for a tangent from an external point to a full circle).

When the centre IS known, a tangent from a point P on the circumference is simply the line through P perpendicular to radius OP (Unit 10, Theorem 2). A tangent from a point P outside the circle is constructed by finding the midpoint M of OP, drawing a semicircle on diameter OP centred at M, and joining P to the point T where this semicircle crosses the given circle — since angle OTP is a right angle (angle in a semicircle), OT is perpendicular to PT, making PT tangent to the circle at T. Two tangents meeting at a specified angle can also be constructed by working backward from the desired angle at the centre.

13.4 Common Tangents to Two Circles, and Circles Touching Lines

Direct (external) common tangents to two EQUAL circles are constructed by drawing parallel diameters perpendicular to the line joining the centres, then joining corresponding endpoints. Transverse (internal) common tangents to two equal circles use a midpoint construction with an auxiliary circle to locate the tangent points on each circle. For two UNEQUAL circles (radii r and r'), direct common tangents use an auxiliary circle of radius r minus r' centred at the larger circle's centre, while transverse common tangents use an auxiliary circle of radius r plus r' — in both cases combined with a semicircle on the line joining the centres to locate the tangent line's direction, then translated by parallels to the second circle.

For two circles that already touch each other (internally or externally), the single common tangent at the point of contact is simply the line perpendicular to the line joining the two centres at that point (Unit 10, Theorem 2). For two intersecting circles, a tangent construction uses an auxiliary circle of radius r minus r-one together with a midpoint circle, similar in spirit to the unequal-circles case. Finally, a circle can be constructed to touch both arms of a given angle by bisecting the angle (any point on the bisector is equidistant from both arms) and dropping a perpendicular from a chosen point on the bisector to one arm for the radius; the same idea extends to a circle touching two converging lines and passing through a specified point between them. A circle touching three converging (non-parallel, non-concurrent) lines is not generally possible, since three lines in general position do not share a single point equidistant from all three in the required tangential sense.

Important Definitions

Key Facts and Relations

Topic	Key Fact / Relation
Circumcircle	Passes through all 3 vertices; centre = intersection of perpendicular bisectors of 2 sides
Incircle	Touches all 3 sides internally; centre = intersection of 2 interior angle bisectors
Escribed circle (e-circle)	Touches 1 side externally, 2 sides extended; centre = intersection of 2 exterior angle bisectors
Inscribed square in a circle	Join the 4 endpoints of 2 perpendicular diameters directly
Inscribed regular hexagon in a circle	6 points spaced 60 degrees apart, found by stepping off chords equal to the radius
Tangent from a point on the circumference	Line through that point, perpendicular to the radius drawn to it
Tangent from a point P outside a circle (centre O)	Semicircle on diameter OP locates the point of tangency T
Direct (external) common tangent, unequal circles (r > r')	Uses an auxiliary circle of radius r – r'
Transverse (internal) common tangent, unequal circles	Uses an auxiliary circle of radius r + r'
Tangent at the point of contact of two touching circles	Perpendicular to the line joining the two centres at that point

Diagrams

Circumcircle and Incircle of a Triangle: A triangle ABC showing its circumcircle (through all three vertices, centre found from perpendicular bisectors of the sides) and its incircle (touching all three sides, centre found from the angle bisectors)

Square Inscribed in a Circle: A circle with centre O and two perpendicular diameters AC and BD, showing the inscribed square ABCD formed by joining the four endpoints in order

Tangent from an External Point (Construction Method): A circle with centre O and external point P, showing the auxiliary semicircle drawn on diameter OP with centre M, meeting the given circle at T, so that PT is the required tangent

Short Questions & Answers

Long Questions & Answers

Describe how to construct the circumcircle, incircle, and escribed circle of a given triangle.

How is the circumcircle of a triangle constructed?

Draw the perpendicular bisectors of any two sides of the triangle; they intersect at the circumcentre O, which is equidistant from all three vertices, giving the circumradius OA = OB = OC.

How is the incircle of a triangle constructed?

Bisect two of the triangle's interior angles; the bisectors meet at the incentre O. Dropping a perpendicular from O to any one side gives the in-radius, and the circle with this radius touches all three sides internally.

How is the escribed circle (e-circle) of a triangle constructed?

Produce two sides of the triangle and bisect the exterior angles formed; these bisectors meet at the e-centre. A perpendicular dropped from the e-centre to the third (un-produced) side gives the e-radius, and the resulting circle touches that side externally and the other two sides extended.

What is the key geometric idea shared by all three constructions?

Each relies on locating a point equidistant from the relevant lines or vertices: the circumcentre is equidistant from the three vertices (via perpendicular bisectors), while the incentre and e-centre are each equidistant from the three sides (via angle bisectors, since every point on an angle bisector is equidistant from the two arms of that angle).

Describe how to construct a tangent from a point outside a circle, and how to construct direct and transverse common tangents to two circles.

How is a tangent constructed from a point P outside a circle with known centre O?

Find the midpoint M of segment OP, then draw a semicircle on OP as diameter with centre M. This semicircle intersects the given circle at a point T; joining P to T gives the required tangent, since angle OTP is 90 degrees (angle in a semicircle), making OT perpendicular to PT.

How are direct (external) common tangents constructed for two equal circles?

Draw diameters of both circles perpendicular to the line joining their centres; joining the corresponding endpoints of these diameters on the same side gives the two direct common tangents.

How are direct common tangents constructed for two UNEQUAL circles?

An auxiliary circle of radius equal to the difference of the two radii is drawn centred at the larger circle's centre; combined with a semicircle on the line joining the centres, this locates points that, when joined through parallels, give the required tangent lines.

How do transverse (internal) common tangents differ in construction from direct tangents?

Transverse tangents cross between the two circles, so their auxiliary circle uses the SUM of the two radii (rather than the difference used for direct tangents), reflecting that the tangent line must pass to the opposite side of each circle.

What construction applies when the two circles already touch each other?

Only one common tangent exists at the point of contact, and it is simply the line perpendicular to the segment joining the two centres, drawn at that point of contact.

Multiple Choice Questions (MCQs)

Quick Revision Summary

• Locate a circle's centre: draw two chords, construct their perpendicular bisectors, intersection = centre
• Circle through 3 non-collinear points: perpendicular bisectors of 2 connecting segments meet at the centre
• Circumcircle: through all 3 vertices; circumcentre = intersection of perpendicular bisectors of 2 sides
• Incircle: touches all 3 sides; incentre = intersection of 2 interior angle bisectors
• Escribed circle: touches 1 side externally, 2 sides extended; e-centre = intersection of 2 exterior angle bisectors
• Inscribed square: join the 4 endpoints of 2 perpendicular diameters
• Inscribed/circumscribed hexagon: 6 points spaced 60 degrees apart (chord = radius gives 60 degree central angle)
• Tangent from a point on the circle: perpendicular to the radius at that point
• Tangent from an external point P: semicircle on OP as diameter locates the tangent point T
• Direct common tangents: don't cross the centres' line | Transverse common tangents: cross between the circles
• Unequal circles: direct tangents use radius (r – r'); transverse tangents use radius (r + r') for the auxiliary circle
• A circle generally cannot be drawn touching three converging (non-concurrent) lines

Exam Tips

• Every construction in this unit reduces to one of two ideas: perpendicular bisectors locate points equidistant from two POINTS, and angle bisectors locate points equidistant from two LINES
• For circumcircle/incircle/escribed-circle questions, identify first whether you need vertices (perpendicular bisectors) or sides (angle bisectors) to be equidistant from
• The semicircle-on-OP trick for tangent construction works because of the angle-in-a-semicircle theorem from Unit 12 — keep that connection in mind
• For common-tangent constructions, remember: DIRECT tangents use the DIFFERENCE of radii, TRANSVERSE tangents use the SUM of radii, for the auxiliary circle
• When two circles already touch, there is only one tangent at the point of contact — perpendicular to the line joining the centres
• Always construct accurately with compass and straightedge and label every intersection point clearly — construction questions are graded on the steps as much as the final figure