A line can pass through a circle in three ways- it can either pass through only a single point of the circle, intersect it, or it can pass outside it. A line passing through a single point of a circle is called a tangent. When a line intersects the circle at two points, we call that line a tangent. Architects and designers use this concept every day in their work. In such cases, it becomes essential to know how to construct a tangent to a circle from a point that is outside the circle. Let’s look at that procedure in detail. 

Tangent

The tangent of a circle is a line that intersects the circle at a single point. The figure given below shows an example of a tangent passing through a circle. The point at which the tangent intersects the circle is called the point of contact. Notice that the tangent is also perpendicular to the line joining the point of contact with the circle. 

Note: The lines which intersect circles at two points are called secants. 

Construction of Tangent

Two tangents can be drawn from any point outside the circle. Our goal is to learn how to construct a single tangent from any point that is given outside the circle. The other tangent can be drawn in a similar manner. 

Steps of Construction: 

We are given a circle with centre O with a radius “r” and a point A outside the circle. Let’s see how to construct a tangent from point A to the circle.

Step 1.  Draw a circle with radius r and let’s call its centre O. Draw a point A outside the circle. 

Step 2. Join OA and bisect it. We’ll call its midpoint M. 

Step 3. Now with M as centre and radius MA. Draw a circle that intersects the given circle at points B and C. 

Step 4. Join AB and AC. They are our given tangents. 

Sample Problems

Question 1: Draw a tangent from a point P which is 10cm from the centre O of the circle of radius 5cm. 

Solution: 

Circle with centre O with a radius “5cm” and a point P outside the circle.

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Step 1.  Draw a circle with radius 5cm and let’s call its centre O. Draw a point A which is 10cm from the centre outside the circle. 

Step 2. Join OP and bisect it. We’ll call its midpoint M. 

Step 3. Now with M as centre and radius MP. Draw a circle that intersects the given circle at points Q and R. 

Step 4. Join PQ and PR. They are our given tangents. 

Question 2: Draw a tangent from a point P which 6cm from the centre O of the circle of radius 3cm. 

Solution: 

Circle with centre O with a radius “3cm” and a point P outside the circle.

Step 1.  Draw a circle with radius 3cm and let’s call its centre O. Draw a point A which is 6cm from the centre outside the circle. 

Step 2. Join OP and bisect it. We’ll call its midpoint M. 

Step 3. Now with M as centre and radius MP. Draw a circle that intersects the given circle at points Q and R. 

Step 4. Join PQ and PR. They are our given tangents. 

Question 3: How many parallel tangents are possible in a circle?

Answer: 

There are at max two possible parallel tangents in a circle. They lie on diametrically opposite points. 

Question 4: Draw a tangent from a point P which 15cm from the centre O of the circle of radius 5cm. 

Solution: 

Circle with centre O with a radius “15cm” and a point P outside the circle.

Step 1.  Draw a circle with radius 5cm and let’s call its centre O. Draw a point A which is 15cm from the centre outside the circle. 

Step 2. Join OP and bisect it. We’ll call its midpoint M. 

Step 3. Now with M as centre and radius MP. Draw a circle that intersects the given circle at points Q and R. 

Step 4. Join PQ and PR. They are our given tangents. 

Question 5: Draw tangents from a point P inclined at 30° from the line joining the point and the centre of the circle of radius 8cm. 

Solution: 

Circle with centre O with a radius “8cm” and a point P outside the circle.

We know that the tangents are inclined at 30°. Let’s say they intersect the circle at R and Q. It makes a triangle OPR. We need to know the length of OP. We also know that OR is perpendicular to PR. Thus, it’s a right-angled triangle, and we can use trigonometric formulas to find out OP.  

[Tex]sin(30) = \frac{OR}{OP} \\ \frac{1}{2} = \frac{OR}{OP}\\ OP = 2 \times OR \\ OP = 2 \times 8 \\ OP = 16[/Tex]

Thus, OP is 16cm. Now we can construct a tangent. 

Step 1.  Draw a circle with radius 8cm and let’s call its centre O. Draw a point P which is 16cm from the centre outside the circle. 

Step 2. Join OP and bisect it. We’ll call its midpoint M. 

Step 3. Now with M as centre and radius MP. Draw a circle that intersects the given circle at points Q and R. 

Step 4. Join PQ and PR. They are our given tangents. 

Question 6: Draw a tangent from a point P which 11cm from the centre O of the circle of radius 6cm. 

Solution: 

Circle with centre O with a radius “6cm” and a point P outside the circle.

Step 1.  Draw a circle with radius 6cm and let’s call its centre O. Draw a point A which is 11cm from the centre outside the circle. 

Step 2. Join OP and bisect it. We’ll call its midpoint M. 

Step 3. Now with M as centre and radius MP. Draw a circle that intersects the given circle at points Q and R.

Step 4. Join PQ and PR. They are our given tangents. 

Practice Questions on Constructions of Tangents to a Circle

1. Construct a circle with radius 4 cm and draw a tangent at any point on the circle.

2. Draw a circle with diameter 10 cm. Construct tangents to this circle from a point 8 cm away from the center.

3. Construct two circles with radii 3 cm and 5 cm, with their centers 10 cm apart. Draw all possible common tangents.

4. Draw a circle with radius 6 cm. Construct a tangent to this circle that is parallel to a given line.

5. Construct a circle that touches two intersecting lines and has its center on the angle bisector of these lines.

6. Draw two concentric circles with radii 3 cm and 5 cm. Construct a tangent to the inner circle that intersects the outer circle at two points.

7. Construct a circle with radius 4 cm. Draw a line segment 12 cm long that is tangent to the circle at one end.

8. Draw two circles with radii 2 cm and 4 cm, with their centers 8 cm apart. Construct the direct common tangents.

9. Construct a circle with radius 5 cm. Draw two parallel tangents to this circle.

10. Draw a circle and a line not intersecting it. Construct tangents to the circle that are perpendicular to the given line.

Summary

Constructing tangents to a circle involves drawing lines that touch the circle at exactly one point, called the point of tangency. These constructions utilize properties of circles, perpendicular lines, and right triangles. Key methods include constructing a tangent from a point outside the circle, drawing tangents at a point on the circle, and creating common tangents to two circles. These constructions are fundamental in geometry and have practical applications in fields like engineering and design.

FAQs on Constructions of Tangents to a Circle

What is a tangent to a circle?

A tangent is a line that touches the circle at exactly one point, called the point of tangency.

How many tangents can be drawn to a circle from an external point?

Two tangents can be drawn to a circle from any external point.

What is the relationship between a tangent and the radius at the point of tangency?

The tangent line is always perpendicular to the radius drawn to the point of tangency.

Can a tangent be drawn from a point inside the circle?

No, tangents can only be drawn from points on or outside the circle.

What are common tangents to two circles?

Common tangents are lines that are tangent to both circles simultaneously. Two circles can have up to four common tangents, depending on their relative positions.