What are the Basic Points About the Tangent?

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Generally, any curved shape can have a tangent. A line that passes outside the circle after touching the circumference of the circle or any curved shape at any single point is said to be a tangent. The point of tangency is the term used for the meeting point of the tangent and the curved surface.  Tangent is a Latin word and the meaning is ‘to touch’.

Defining: the tangent of a circle

Any line that intersects with the circumference of the circle at one point is termed to be a tangent. The common meeting point of the tangent and the circumference is said to be the point of tangency.

Tangent touches the circle only at one point, unlike the secant that touches the circumference of the circle at two different points. This means a tangent is always seen or constructed at the outer edge of the circle while a secant is drawn inside the circle.

What are the properties of tangent?

Cuemath explains the concept of properties and theorems of the Circles with related examples that are easy to understand. Let’s go through the properties of the tangent thoroughly.

  • In a circle tangent never touches more than one point on the circumference.
  • The radius of the respective circle and tangent of the same are always perpendicular to each other which means they meet at right angles i.e., 90 degrees.
  • A tangent can never pass through the interior of the circular figure. This property means that a tangent never cuts the circle into two parts.
  • If two tangents are drawn from a common exterior point on the circumference of the same circle then both have equal distance between the point and the point of tangency i.e., both have an equal measurement.
  • A special case of tangent is that two circles can be a tangent to one other but only if they touch each other at only one point does that mean they should not cut each other.

The properties of the tangent make the concept of tangent clear and precise.

The tangent theorem of a circle

The theorem explains the relation between the radius of the circle and the tangent drawn to a circle in respect to point of tangency. It says that any straight line stretched from any exterior point on the circumference of the circle can be referred to as a tangent only when it from a right angle with the radius of the circle at the respective point of tangency.

The widely accepted method to find the measure of the tangent

Let’s assume that a straight-line BD (which cuts the circle at points B and C) forms the secant of a circle and a straight line say, AB is the tangent of the same circle.

Then the relation becomes

BD/AB=AB/CD, cross-multiplying the denominators we get, BD*CD=AB^2and this gives the general equation of the tangent.

This relation makes the calculations easy and simple. Let’s understand the same with the help of an example so that we may have clarification of the topic.

Example: Consider a circle with PQR as a secant of the circle cutting the circle at points P and Q where PQ=10 cm while QR is 18 cm. Now taking R as the on-point among secant and tangent drop o line on the circumference showing the point of tangency as point S.

Then what will be the length of the tangent PS?


Given that PQ is 10 cm.

QR=18 cm.

So, we get PR is 10+18 =28 cm.

Now, according to the above-mentioned relation SR^2=PR*PQ

I.e., SR^2=28*18

Now, SR=√504

And SR= 22.4 cm.

Example: Let’s consider one more example.

If the radius of any circle is 6 cm and the secant is 10 cm. What is the length of the tangent drawn from the same point from which the secant has been drawn?

Solution: Since we have already studied above that the radius is perpendicular to the tangent at the point of tangency so all the three ll together make a right-angled triangle. Hence applying the Pythagoras theorem,

We get, tangent^2= Secant^2-radius^2

I.e., Tangent= √10^2-6^2= √100- 36=√64= 8 cm.

So, the tangent is 8cm. in length.

Cuemath’s online math app describes the concepts with examples in an efficient manner. In the present-day world, online math teaching has been a feature of perfect understanding and clarity of concepts.

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