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Tangents to a Circle

Tangent to a circle is a line that touches the circle at one point, which is known as Tangency. At the point of Tangency, Tangent to a circle is always perpendicular to the radius. Let us learn more about tangents in this chapter.

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Tangent to a Circle

The line that joins two infinitely close points from a point on the circle is a Tangent. In other words, we can say that the lines that intersect the circles exactly in one single point are Tangents. Point of tangency is the point where the tangent touches the circle. At the point of tangency, a tangent is perpendicular to the radius. Several theorems are related to this because it plays a significant role in geometrical constructions and proofs. We will look at them one by one.

tangent to a circle

[Source: Illustrative Mathematics]

Tangents Formula

The formula for the tangent is given below. For the description of formula, please look at the following diagram.

tangent to a circle

[Source: Jim Wilson at UGA]

Here, we consider a circle where P is the exterior point. From that exterior point, the circle has the tangent at a points A and B. A straight line which cuts curve into two or more parts is known as a secant. So, here secant is PR is drawn and at Q, R intersects the circle as shown in the upper diagram. The formula for tangent-secant states that:

PR/PS = PS/PQ

PS= PQ.PR

Properties of Tangents

Remember the following points about the properties of tangents-

  • The tangent line never crosses the circle, it just touches the circle.
  • At the point of tangency, it is perpendicular to the radius.
  • A chord and tangent form an angle and this angle is same as that of tangent inscribed on the opposite side of the chord.
  • From the same external point, the tangent segments to a circle are equal.

Learn more about Arc of a Circle here in detail.

Download Arc of a Circle Cheat Sheet PDF

circle cheat sheet

Theorems for Tangents to Circle

Theorem 1

A radius is obtained by joining the centre and the point of tangency. The tangent at a point on a circle is at right angles to this radius. Just follow this below diagram: Here AB⊥OP

tangent to a circle

[Source: Gradeup]

Theorem 2

This theorem states that if from one external point, two tangents are drawn to a circle then they have equal tangent segments. Tangent segment means line joining to the external point and the point of tangency. Consider the following diagram: Here, AC=BC.

tangent to a circle

[Source: Mathematics Stack Exchange]

These are some of the basic theorems on tangents to a circle. In the next section, we will look at some of the basic properties of tangents.

Browse more Topics Under Circles

Learn more about Cyclic Quadrilateral here in detail.

Solved Example for You

Question 1: Give some properties of tangents to a circle.

Answer: The properties are as follows:

  • The tangent line never crosses the circle, it just touches the circle.
  • At the point of tangency, it is perpendicular to the radius.
  • A chord and tangent form an angle and this angle is the same as that of tangent inscribed on the opposite side of the chord.
  • From the same external point, the tangent segments to a circle are equal.

Question 2: What is the importance of a tangent?

Answer: The tangent line is valuable and necessary because it permits us to find out the slope of a curved function at a specific point at the curve. We have learned that we are able to find the slope of a line, but we’ve never got to know how to find out the slope in a curved function.

Question 3: What does it mean when the lines are tangent?

Answer: If any line is touching a curve at a point and not even crossing over, then, it is a line that crosses a differentiable curve at some point where the slope of that curve equals the slope of the line. In addition, a line that is tangent to a circle forms a perpendicular at the radius to the point of tangency.

Question 4: How can we draw a tangent?

Answer: Point to Tangents on a Circle:

  1. Make a line that connects the point to the middle of the circle.
  2. Draw the perpendicular bisector for that line.
  3. Locate the compass on the centre, adjust its length to reach till the end-point, and then, make an arc through the circle.
  4. Finally, where the arc crosses the circle will be known as the tangent point.

Question 5: How do we know if 2 circles are tangent?

Answer: When a line is a tangent to a circle, then it states that the line is touching the circle exactly at a single point. A circle can be tangent to the other circle, it means that the 2 circles are touching exactly at one point.

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