Definition Theorem And Steps To Construct Tangent To A 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 In Geometry
1. What is the construction of a tangent to a circle?
The construction of a tangent to a circle is the geometric method of drawing a line that touches the circle at exactly one point without cutting it. A tangent meets the circle at only one point called the point of contact.
- At the point of contact, the tangent is perpendicular to the radius.
- This property is the key rule used in geometric constructions.
- Tangents can be constructed from a point on the circle or from a point outside the circle.
2. How do you construct a tangent to a circle at a given point on the circle?
To construct a tangent at a point on a circle, draw a line perpendicular to the radius at that point. Steps:
- Join the centre O of the circle to the given point P on the circle.
- Construct a line perpendicular to OP at point P.
- The perpendicular line is the required tangent at P.
3. How do you construct tangents to a circle from a point outside the circle?
To construct tangents from an external point, use the midpoint method with the line joining the point and the centre. Steps:
- Join the external point P to the centre O.
- Find the midpoint M of OP.
- With centre M and radius MO, draw a circle.
- This circle intersects the given circle at points T₁ and T₂.
- Join P to T₁ and P to T₂.
4. Why is the tangent perpendicular to the radius at the point of contact?
A tangent is perpendicular to the radius at the point of contact because any other line would intersect the circle at two points. The radius drawn to the point of contact makes a 90° angle with the tangent.
- If the angle were not 90°, the line would pass inside the circle.
- Hence, perpendicularity ensures the line touches the circle at exactly one point.
5. How many tangents can be drawn from a point to a circle?
From a point outside a circle, exactly two tangents can be drawn; from a point on the circle, only one tangent can be drawn. From a point inside the circle, no tangent can be drawn.
- External point → 2 tangents
- On the circle → 1 tangent
- Inside the circle → 0 tangents
6. What is the formula for the length of a tangent from an external point?
The length of a tangent from an external point is given by PT = √(OP² − r²). Here:
- OP = distance from external point to centre
- r = radius of the circle
- PT = length of the tangent
7. Can you give an example of finding the length of a tangent?
Yes, the length of a tangent can be calculated using the formula PT = √(OP² − r²). Example:
- Let OP = 13 cm and r = 5 cm.
- PT = √(13² − 5²)
- PT = √(169 − 25)
- PT = √144 = 12 cm
8. What are the properties of tangents to a circle?
The main properties of tangents to a circle include perpendicularity and equal lengths from an external point. Key properties:
- A tangent touches the circle at exactly one point.
- The tangent at any point is perpendicular to the radius at that point.
- Tangents drawn from an external point are equal in length.
9. What is the difference between a tangent and a secant of a circle?
A tangent touches the circle at one point, while a secant cuts the circle at two points. Differences:
- Tangent → 1 common point
- Secant → 2 intersection points
- Tangent is perpendicular to radius at contact point
- Secant is not necessarily perpendicular to the radius
10. What are common mistakes in constructing a tangent to a circle?
Common mistakes in tangent construction include not drawing the perpendicular correctly and misidentifying the centre. Typical errors:
- Not ensuring the tangent is perpendicular to the radius.
- Incorrect midpoint when constructing tangents from an external point.
- Confusing a secant with a tangent.
