 # Construction of Tangent to a Circle

A tangent of a circle is a line that starts from outside the circle and intersects the plane of the circle at its periphery at one exact point. The point where a tangent intersects the circle is called the point of tangency (See Figure 1). Construction of Tangent is one of the most basic parts of geometry. Let’s study more.

### Suggested Videos        ## Tangents

A common tangent can be any line, a ray or a segment that can be a tangent to more than one circle at the same time. There can be one to four number of common tangents to two circles. The point to be noted is that a tangent of a circle touches the circle but never enters it. Also, multiple tangents to the two circles can be an outer tangent or an inner one to each other’s circles. Figure 1: Circle with a centre O and tangent as PQ. X is the point of tangency

Note: There can only be two tangents drawn from one point outside the circle. In this module, we’ll learn the art of constructing two tangents to a given circle from a point outside the circle without the help of a scale.

## Constructing Tangent of a Circle

We can come across two cases while constructing tangent of a circle. These are described as follows:

### Case I: When the centre of a circle is known

Let’s say that we are given a circle with centre O and a point P outside it, and we have to construct two tangents from the point P to the circle without the help of a ruler. Follow the steps of construction as below:

1. Join OP and bisect it.  To bisect OP, take a compass, and open it slightly more than half of the length of the line segment.
2. From point P, mark a minimal arc above and below the line segment. Repeat the similar step from point O keeping the opening of the compass as same as it was from point P. Two points will be created where the two arcs, produced from point O and point P, meet.
3. Joint these two points with a line segment using a scale. This line segment bisects the OP. Let’s consider H as the mid-point of PO.
4. Taking the point H as a centre and HO as a radius, draw a circle. Let it intersect the given circle at the points, T and T’.
5. Join PT and PT’. Then PT and PT’ are the required two tangents. Figure 2: Drawing tangents PT and PT’ (marked in red) to the circle

### Proof and Rationale

To check if the construction is correct:

• Join OT. Then ∠PTO is an angle in the semicircle and, therefore, ∠PTO = 90° [ angle in a semicircle is always 90°]
• Check with the 180° protractor ensuring that ∠PTO = 90°.

Since OT is the radius of the circle with centre O and ∠PTO = 90°, therefore, PT has to be a tangent to the circle. This is because of the property of the circle which states that the radius from the centre of the circle to the point of tangency is perpendicular to the tangent line. Similarly, PT’ also becomes a tangent to the circle.

### Case II: When the centre of the given circle is not known

In that case:

1. Draw any two non-parallel chords in the given circle
2. Draw perpendicular bisectors to both of the chords (as described in steps 2 and 3 above in Case I) (See Figure 3)
3. Find the point of intersection of two perpendicular bisectors which will be the centre of the given circle.
4. Follow the same steps from 1 to 5 as described above in Case I. Figure 3: Intersection of two perpendicular bisectors is the centre point O

## Solved Examples for You

Question 1: Given a circle without a centre point and a point P outside the circle. Find the centre point of the circle and draw two tangents from the point P to the circle. Describe the steps.

1. Draw any two non-parallel chords (CD and EF) in the given circle
2. Draw perpendicular bisectors to both of the chords.
1. To bisect CD, take a compass, and open it slightly more than half of the length of the line segment.
2. From point D, mark a minimal arc above and below the line segment. Repeat the similar step from point C keeping the opening of the compass as same as it was from point D. Two points will be created where the two arcs, produced from point C and point D, meet.
3. Joint these two points with a line segment using a protractor. This line segment bisects the CD.
4. Repeat steps from 2 a to 2 c for drawing a perpendicular bisector for EF.
3. Find the point of intersection of two perpendicular bisectors which will be the centre of the given circle, say point O. (See Figure A)
4. Join the centre point O to the given point P outside the circle.
5. Bisect OP by following steps from 2 a to 2 c. Taking the point G as a centre and OG as a radius, draw a circle. Let it intersect the given circle at the points, T and T’.
6. Join PT and PT’. Then PT and PT’ are the required two tangents.

Question 2: What does it mean if a line is tangent to a circle?

Answer: A tangent line refers to a line that is intersecting a circle at one point. We call such a line the tangent to that circle. Moreover, the point at which the circle and the line intersect is referred to as the point of tangency. In other words, for any tangent line, there will be a perpendicular radius

Question 3: How many tangents can a circle have?

Answer: A circle has 0 tangents. We can draw exactly 0 tangents through a circle. A tangent will intersect a circle in only one point so that it does not enter the area of the circle. Moreover, we can draw an infinite number of tangents to a certain circle as there won’t be any limits to the points for a tangent to intersect.

Question 4: Is a tangent line perpendicular?

Answer: A tangent to a circle is a line that intersects the circle at precisely one point. It is the point of tangency or tangency point. A significant result is that the radius from the centre of the circle to the point of tangency will be perpendicular to the tangent line

Question 5: What does tangent mean?

Answer: In geometry, the tangent line or sometimes simply called the tangent, is a plane curve at a certain point is the straight line which “just touches” the curve at that point. Leibniz stated that it is the line through a pair of infinitely close points on the curve. Moreover, the word “tangent” is derived from the Latin ‘tangere’ which means ‘to touch’. Figure A: Tangents PT and PT’ from the given point P outside the given circle (highlighted)

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