10. CIRCLES

You have studied in Class IX that a circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). You have also studied various terms related to a circle like chord, segment, sector, arc etc. Let us now examine the different situations that can arise when a circle and a line are given in a plane.

A circle is defined as a collection of points that are all the same distance away from a single point, known as the center. The distance from the center to any point on the circle is referred to as the radius. Additionally, terms like chord (a line segment connecting two points on a circle), segment (part of a circle cut off by a chord), sector (part of a circle enclosed by two radii), and arc (part of the circumference) are important in understanding circles. This section will explore how a line can interact with a circle in different ways.

Imagine you are in a playground with a merry-go-round. If you stand still and place a toy at a distance from the center of the merry-go-round, that toy marks a circle. As the ride spins, you can visualize the relationship between the center and the circle created by the toy.

So, let us consider a circle and a line PQ. There can be three possibilities: (i) the line PQ and the circle have no common point, (ii) there are two common points, and (iii) there is only one point common to the line and the circle.

When a line is drawn with respect to a circle, there are three possible relationships: first, the line does not touch the circle at all, which means it is a non-intersecting line. Second, if the line crosses the circle at two points, we call it a secant. Lastly, if the line just touches the circle at one point, it is termed a tangent. Each of these interactions is critical for understanding the geometry of circles.

Think of a hula hoop. If a straight stick (line) is moved and never touches the hoop, that's a non-intersecting line. If the stick pokes through the hoop at two points, that’s like a secant. Finally, if you manage to rest the stick against the side of the hoop without crossing, that's akin to a tangent.

In the previous section, you have seen that a tangent to a circle is a line that intersects the circle at only one point.

A tangent line is unique in that it meets the circle at exactly one point. This characteristic distinguishes it from a secant, which intersects the circle at multiple points. The tangent can be visualized as merely 'touching' the circle without crossing it, hence the name derived from the Latin word 'tangere', meaning 'to touch'.

Picture a skateboard wheel on a ramp: the edge of the ramp at the point of contact with the wheel acts as a tangent, just touching the wheel at a single point without going inside the wheel.

Activity 1: Take a circular wire and attach a straight wire AB at a point P of the circular wire so that it can rotate about the point P. Rotate the wire AB to observe how it intersects the circular wire. You will find that there is only one tangent at a point of the circle.

This activity helps to visualize how tangents work. By physically moving a line around a circular structure, students can observe that there is a specific moment when the line just touches the circle, confirming that there is only one tangent that exists at any given point on the circular wire.

Imagine a clock hand that moves around the clock face. When the hand points directly at any number, it 'touches' the clock face at that number—it doesn’t go through the clock but just meets it right there.

The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.

This explains the relationship between secants and tangents. As a secant is defined by two points intersecting the circle, when these points merge into one, it simplifies into a tangent. This property signifies a unique association between the two lines and how they interact with the circle.

Imagine two people crossing paths at a park: their intersection forms a secant path. If one person stops walking while the other continues on the circle’s circumference, they end up creating a tangent interaction at that stopping point.

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

This theorem denotes that at any tangent point on a circle, the radius drawn to that point will always form a right angle with the tangent. This reinforces the geometric nature of circles and allows for further deductions regarding angles and other tangent properties within the circle.

Visualize a door touching the wall at a right angle—the hinges represent the radius, while the edge of the door represents the tangent, perfectly illustrating how the door only touches the wall at one specific point.

To summarize, from an external point to a circle, there can be two tangents, one tangent from the point of contact, or no tangible connection at all depending on the location of the external point.

This concept is fundamental for understanding the tangent lengths from points relative to circles. Depending on whether the point is inside, on the edge, or outside of the circle, the number of tangents varies. The relationships formed can help solve real-world geometric problems regarding circles.

Consider a flashlight beam directed at a round object (like a basketball). Depending on where the flashlight is positioned (inside, on, or outside the object), you’ll either get light hitting it at multiple angles, just touching it, or not touching it at all.

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