Circles
This chapter covers the tangent to a circle, the key property that a tangent is perpendicular to the radius at the point of contact, and the lengths of tangents drawn from an external point.
A tangent to a circle is a line that touches the circle at exactly one point, called the point of contact or point of tangency. Unlike a secant, which intersects the circle at two points, a tangent touches it only once.
The tangent at any point of a circle is perpendicular to the radius through the point of contact. This is the most important and frequently used property in this chapter for solving geometric problems involving tangents.
Q. How many tangents can be drawn to a circle from a point lying outside the circle?
Exactly two tangents can be drawn to a circle from an external point, and these two tangents are always equal in length.
From a point on the circle, exactly one tangent can be drawn. From a point inside the circle, no tangent can be drawn. From a point outside the circle, exactly two tangents can be drawn.
Q. A circle has a radius of 5 cm. Find the length of a tangent drawn from a point 13 cm away from the centre.
Since the tangent is perpendicular to the radius at the point of contact, the radius, tangent, and the line from the external point to the centre form a right triangle, with the line to the centre as the hypotenuse. Length of tangent = root(13^2 - 5^2) = root(169 - 25) = root(144) = 12 cm.
The angle between a tangent to a circle and the radius drawn at the point of contact is:
Correct answer: (C) 90 degrees
The two tangents drawn from an external point to a circle are always _______ in length.
Equal
A line intersecting a circle at two points is called a tangent.
False — a line intersecting a circle at two points is called a secant. A tangent touches the circle at exactly one point.