Construction Tangent Circle
A tangent to a circle is a line that is perpendicular to the radius at a particular point. The point where the radius and tangent are perpendicular to each other is known as the point of tangency. There are various conditions and precludes for the construction of tangents to a circle as mentioned below:
At a Particular Point of the Circle with Centre O:
Let’s take a circle with centre O and a point P on its circumference.
Hence, OP will be the radius of the circle.
Extend the radius OP further, outside the circle till M.
Now, adjust the compass in such a way, so that the opening of the compass is more than the radius OP.
Once the compass is adjusted accordingly, cut a semi-circle on OM while keeping the compass on O.
Similarly, cut a semi-circle by keeping the compass at M.
Now, join the semi-circles so formed to draw a tangent to a circle.
The point where OP and the perpendicular meet will be the point of tangency.
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At a Point Outside The Circle:
Let’s take a circle with centre O and point P outside the circle.
Join the points O and P.
Now, construct a perpendicular bisector of the line OP.
For making a perpendicular bisector to OP, adjust the mouth of the compass in such a way that it is more than half of OP.
Now, while putting the compass on point O, construct an arc around the middle of OP.
Similarly, construct an arc with the compass on point P.
Make a line, joining the points where arcs are intersecting.
The line passing through OP will be the perpendicular bisector of it.
Now, construct a circle with O and the point of intersection.
Connect the point P with the two points at which both the circles are intersecting and draw tangent to circle.
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Construction of Tangents When NO Centre is Given:
There are two methods of construction of tangent without using centre:
Inscribed Triangle
Construct a circle and make an inscribed triangle, ABM inside it.
Now keep the compass at point A and measure angle BAM through the compass.
Adjust the mouth of the compass according to the angle.
Keep the compass at point M and cut an angle keeping the opening of the compass same, at the line BM passing through the circle.
Now, adjust the mouth of the compass according to the width of the angle BAM.
With the same mouth opening, keep the compass at the point of intersection the line BM and arc and cut the circle.
The two arcs will be intersecting at a point Q.
Now join the points Q and M, and draw a tangent to the circle without using centre.
This line will be the tangent to the circle.
Chords
Yet another way of construction of tangents in a circle, without centre is by making chords. Construct any 2 chords inside the circle having a common point.
Now, make the perpendicular bisector of both the chords.
To make the perpendicular bisector, keep the compass at one point of a chord and open the mouth of the compass in such a way that it is more than half the length of the chord.
Cut an arc on the chord.
Similarly, put the compass on the other end of the chord and cut an arc.
Join the points and make a line where these two arcs are cutting each other.
It will be the perpendicular bisector of the chord.
Similarly, construct a perpendicular bisector at another chord.
The point where both the perpendicular bisectors will meet will be the centre of the circle.
Now that we get the centre O, we can make a radius OP to the circle.
Extend OP outside the circle till M.
Now, construct a perpendicular bisector of the line OP.
To construct a perpendicular bisector, open the mouth of the compass more than half of the length of OP.
Keeping the compass at point O, cut a semicircle at the line OP, similarly keep the compass at point M make a semi-circle.
Join the lines and draw a tangent without using centre through the points where the semi-circles are intersecting.
The line will be tangent to the circle.
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FAQs on Construction of Tangent to a Circle
1. What is a tangent to a circle in geometry?
In geometry, a tangent is a straight line that lies in the same plane as a circle and touches the circle at exactly one point. This single point where the line touches the circle is known as the point of tangency. Unlike a secant which cuts through the circle at two points, a tangent only grazes its edge.
2. What are the steps to construct a tangent to a circle from an external point?
To construct a tangent to a circle from a point outside it, you need a compass and a straightedge. The primary steps as per the CBSE curriculum are:
- Connect the centre of the circle (O) to the external point (P).
- Construct the perpendicular bisector of the line segment OP to find its midpoint (M).
- With M as the centre and MO (or MP) as the radius, draw a new circle.
- This new circle will intersect the original circle at two points. Let's call them T1 and T2.
- Join the external point P to T1 and T2. The lines PT1 and PT2 are the two required tangents to the circle.
3. Why is the line joining the external point and the centre bisected during this construction?
Bisecting the line segment OP (from the external point P to the centre O) is a critical step to locate the points of tangency. When a new circle is drawn with OP as the diameter, any angle inscribed in this semicircle is a right angle (90°). This ensures that the angle formed by the radius and the constructed tangent (e.g., ∠OT1P) is exactly 90°, which is the fundamental property of a tangent.
4. How many tangents can be drawn to a circle from a single point?
The number of tangents that can be drawn from a point depends on its position relative to the circle:
- From a point inside the circle: Zero tangents can be drawn.
- From a point on the circle: Exactly one unique tangent can be drawn.
- From a point outside the circle: Exactly two tangents can be drawn.
5. What is the relationship between a circle's radius and its tangent at the point of tangency?
The radius of a circle is always perpendicular to the tangent at the point of tangency. This means that the angle between the radius (drawn to the point of contact) and the tangent line is exactly 90 degrees. This theorem is the geometric basis for the construction of tangents.
6. Is it possible to construct a tangent to a circle without using its centre?
Yes, it is possible to construct a tangent without knowing the circle's centre. This advanced construction method uses the properties of chords and secants. One common method involves drawing a secant from the external point P that intersects the circle at points A and B. Then, using geometric relationships (like the tangent-secant theorem), the tangent can be constructed without any reference to the circle's centre.
7. What is the key difference between a tangent and a secant of a circle?
The key difference lies in the number of intersection points. A tangent is a line that touches a circle at precisely one point. In contrast, a secant is a line that cuts through a circle, intersecting it at two distinct points. A chord is simply the line segment of a secant that lies inside the circle.
8. What geometric tools are essential for the construction of a tangent?
For performing geometric constructions of tangents as per the NCERT syllabus for Class 10, the essential tools required are:
- A ruler or a straightedge for drawing straight lines.
- A pair of compasses for drawing arcs and circles.
- A sharp pencil for accurate markings.
A protractor can also be useful for verifying that the angle between the radius and the tangent is 90 degrees, but it is not typically used for the primary construction steps.
