Hello everyone, and welcome to today's mathematics lesson. We are going to explore constructions involving circles. By the end of this session, you will understand how to construct tangents to circles, how to draw circumscribed and inscribed circles for triangles, and how to work with circles related to regular hexagons. Let us begin.
First, let us understand what a tangent to a circle actually is. A tangent is a straight line that touches the circle at exactly one point, called the point of contact. Here is a fundamental property you must always remember: the angle between the radius and the tangent at the point of contact is always 90 degrees. This property forms the basis of all our constructions.
Construction of a tangent through a point on the circumference. Suppose you have a circle with centre O, and P is a point lying on the circle itself. Here is how you construct the tangent at P. First, join the centre O to the point P. Then, construct a line through P that makes a 90-degree angle with OP. This line is your required tangent. The construction relies entirely on the fact that ∠OPA = 90°, ensuring the line is perpendicular to the radius at the point of contact.
Now, what if the point lies outside the circle? Construction of tangents from an exterior point. Let P be a point outside the circle with centre O. Join P to O. Now, here is the clever step: draw a new circle with OP as its diameter. This new circle will cut the original circle at two points, say A and B. Join PA and PB. These are your two tangents from the external point P.
Why does this work? Since OP is the diameter of the auxiliary circle, any angle subtended by it on the circumference is 90 degrees. Therefore, ∠PAO = 90° and ∠PBO = 90°. This means PA and PB are both perpendicular to the radii OA and OB respectively, making them tangents.
Two important results follow from this construction. First, the two tangents drawn from an external point to a circle are always equal in length, so PA equals PB. Second, by the Pythagorean theorem, PA² + OA² = OP².
Moving on to triangles and their associated circles. Every triangle has two special circles associated with it: the circumscribed circle and the inscribed circle.
The circumscribed circle, or circumcircle, passes through all three vertices of the triangle. To construct it, draw the perpendicular bisectors of any two sides of the triangle. These bisectors meet at a point called the circumcentre, denoted by O. With O as centre and radius equal to OA, OB, or OC — they are all equal — draw the circle. This circle passes through A, B, and C. The distance from O to any vertex is called the circumradius.
The key principle here is that the perpendicular bisectors of the sides of a triangle are concurrent, meaning they all meet at a single point. This point, the circumcentre, is equidistant from all three vertices of the triangle.
The inscribed circle, or incircle, touches all three sides of the triangle from inside. To construct it, draw the bisectors of any two angles of the triangle. These angle bisectors meet at a point called the incentre, denoted by I. From I, drop a perpendicular to any side, meeting it at point D. With I as centre and ID as radius, draw the circle. This circle touches all three sides of the triangle.
The angle bisectors of a triangle are also concurrent, meeting at the incentre. The incentre is equidistant from all three sides of the triangle, and this distance is called the inradius.
These constructions apply to all triangles, whether equilateral, isosceles, or scalene. The methods remain identical regardless of the triangle's shape.
Now let us extend these ideas to regular hexagons. A regular hexagon has six equal sides and six equal angles. Each interior angle of a regular hexagon equals 120 degrees, calculated using the formula ((2n-4)/n) × 90°, which gives 120 degrees when n equals 6.
To construct a circle circumscribing a regular hexagon, draw the perpendicular bisectors of any two sides. These meet at the centre O. With O as centre and distance to any vertex as radius, draw the circle through all six vertices.
Here is a useful fact. For a regular hexagon, the circumradius always equals the side length of the hexagon. Alternatively, you can find the centre by drawing two main diagonals — the longest diagonals connecting opposite vertices — which intersect at the centre.
To construct an inscribed circle in a regular hexagon, draw the bisectors of any two interior angles. These meet at point I. From I, drop a perpendicular to any side, meeting it at P. With I as centre and IP as radius, draw the circle touching all six sides.
Note carefully. If the hexagon is not regular, these constructions may not be possible. A circle cannot always be drawn to pass through all vertices of an irregular hexagon, nor can a circle always be inscribed to touch all its sides.
Let me now recap the key takeaways from this chapter.
First, a tangent to a circle is perpendicular to the radius at the point of contact. Second, from an external point, two equal tangents can be drawn to a circle. Third, the circumcentre of a triangle is found where perpendicular bisectors of sides meet; it is equidistant from all vertices. Fourth, the incentre is found where angle bisectors meet; it is equidistant from all sides. Fifth, for a regular hexagon, the circumradius equals the side length. Sixth, only regular polygons guarantee both circumscribed and inscribed circles.
That brings us to the end of our lesson on Constructions with Circles. Remember, precision in construction comes from understanding the underlying geometric principles. Always verify your constructions by checking the key properties: perpendicularity for tangents, equal distances for circumcentres and incentres. Keep practising, and you will master these techniques. Thank you for listening, and see you in the next lesson.