Definition of Tangent to Circle
A line that joins two close points from a point on the circle is known as a tangent. In simple words, we can say that the lines that intersect the circle exactly in one single point are tangents. Only one tangent can be at a point to circle. The point where a tangent touches the circle is known as the point of tangency. The point where the circle and the line intersect is perpendicular to the radius. As it plays a vital role in the geometrical construction there are many theorems related to it which we will discuss further in this chapter.
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Here, point O is the radius, point P is the point of tangency.
Various Conditions of Tangency
Only when a line touches the curve at a single point it is considered a tangent. Or else it is considered only to be a line. Hence, we can define tangent based on the point of tangency and its position with respect to the circle.
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When point lies on the circle
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When point lies inside the circle
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When point lies outside the circle
When Point Lies on the Circle
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Here, from the figure, it is stated that there is only one tangent to a circle through a point that lies on the circle.
When Point Lies Inside the Circle
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In the figure above, the point P is inside the circle. Now, all the lines passing through point P are intersecting the circle at two points. therefore, no tangent can be drawn to the circle that passes through a point lying inside the circle.
When Point Lies Outside the Circle
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The above figure concludes that from a point P that lies outside the circle, there are two tangents to a circle.
Properties of Tangent
Always remember the below points about the properties of a tangent
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A line of tangent never crosses the circle or enters it; it only touches the circle.
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The point at which the lien and circle intersect is perpendicular to the radius
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The tangent segment to a circle is equal from the same external point.
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A tangent and a chord forms an angle, the angle is exactly similar to the tangent inscribed on the opposite side of the chord.
Equation of Tangent to a Circle
Below is the equation of tangent to a circle
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Tangent to a circle equation x2+ y2=a2 at (a cos θ, a sin θ) is x cos θ+y sin θ= a
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Tangent to a circle equation x2+ y2=a2 at (x1, y1) is xx1+yy1= a2
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Tangent to a circle equation x2+ y2=a2 for a line y = mx +c is y = mx ± a √[1+ m2]
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Tangent to a circle equation x2+ y2=a2 at (x1, y1) is xx1+yy1= a2
Tangent to a Circle Formula
To understand the formula of the tangent look at the diagram given below.
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Here, we have a circle with P as its exterior point. From the exterior point P the circle has a tangent at Point Q and S. A straight line that cuts the curve in two or more parts is known as a secant. So, here the secant is PR and at point Q, R intersects the circle as shown in the diagram above. So, now we get the formula for tangent-secant
PR/PS = PS/ PQ
PS² = PQ.PR
Theorems of Tangents to Circle
Theorem 1
A radius is gained by joining the centre and the point of tangency. A tangent at the common point on the circle is at a right angle to the radius. The below diagram will explain the same where AB [perp] OP
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Theorem 2
From one external point only two tangents are drawn to a circle that have equal tangent segments. A tangent segment is the line joining to the external point and the point of tangency. According to the below diagram AC = BC
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Examples of a Tangent to a Circle Formula
Example 1
In the below circle point O is the radius, PT is a tangent and OP is the radius, If PT is a tangent, then OP is perpendicular to PT.
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If OP = 3 Units and PT = 4 Units. Find the length of OT
Solution: as the radius is perpendicular to the tangent at the point of tangency, OP [perp] PT
Therefore, ∠P is the right angle in the triangle OPT and triangle OPT is a right angle triangle.
Now, according to the Pythagoras theorem, we find OT.
(OP)² + (PT)² = (OT)²
3² + 4² = (OT)²
9 + 16 = (OT)²
25 = (OT)²
5 = OT
As the length cannot be negative, the length of OT is 5 units.
Example 2
In the below diagram PA and PB are tangents to the circle. Find the value of
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∠OAP
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∠AOB
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∠OBA
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∠ASB
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The length of OP, PB = 7 cm (given)
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Solution:
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∠OAP = 90° (Tangent is perpendicular to the radius)
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∠AOB + ∠APB = 180°
∠AOB + 48° = 180°
∠AOB = 180° – 48° = 132°
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∠OBA + ∠OAB + ∠AOB = 180° (angle sum of triangle)
2 x ∠OBA + ∠AOB = 180° (∠OBA = ∠OAB)
2 x ∠OBA + 132° = 180° (∠AOB = 132°)
∠OBA = 24°
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∠AOB = 2 x ∠ASB (angle at centre = 2 angle at circle)
∠ASB = ∠AOB / 2
∠ASB = 132° / 2 = 66°
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Cos 24° = [frac{7}{OP}] ⇒ OP = [frac{7}{cos24^{0}}]