Length of Tangent on a Circle: Formula and Theorems

A tangent to a circle can be defined as a line that touches the circle exactly at one point. The point of contact is the point at which the tangent touches the circle. Depending on the position of this point, we can consider that the number tangents than can be drawn to a circle. A tangent is defined as a straight line that intersects the circle exactly at one point. Here, we will be discussing more about the formulas, and theorems of the length of tangent on a circle along with some important questions.

Key Terms: Tangent, Tangent of Circle, Tangent Theorems, Pythagoras Theorems, Tangent Formulas, Circle Formulas

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What is Tangent to a Circle?

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Tangent to a circle is defined as the line that touches the circle only at one point. There cannot be more than one tangent at a point to circle. Point of tangency can be defined as the point at which tangent meets the circle.

Tangent to a Circle

The above figure has a circle with centre O. Where the tangent is drawn to a circle through point C. A point D is taken on tangent AB other than C and join OD. The point D which is taken on the tangent AB lies outside the circle because; if point D lies inside the circle, then AB will be a secant to the circle and it will not be a tangent as mentioned earlier.

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Tangent Equation

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These are the following equations of the tangent:

• The tangent to a circle equation x² + y² = a² at (x₁, y₁) is xx₁ + yy₁ = a²
• The tangent to a circle equation x² + y² + 2gx + 2fy + c = 0 at (x₁, y₁) is xx₁ + yy₁ + g(x+x₁) + f(y +y₁) + c = 0
• The tangent to a circle equation x² + y² = a² at (a cos θ, a sin θ) is x cos θ + y sin θ = a
• The 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 Theorems

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Theorem 1

A radius which is obtained by joining the centre and the point of tangency and this tangent at a point on a circle is at right angles to the radius obtained. To understand the statement, follow the diagram below: Here AB⊥OP

Theorem 2

It states that from one external point, two tangents are drawn to a circle then they have equal tangent segments. The meaning of tangent segment is the line joining to the external point and the point of tangency. To understand the statement, follow the diagram below: Here, AC = BC.

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Tangent of Circle Formula

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Suppose a point P lies outside the circle.

From that point P, we draw two tangents to the circle meeting at point E and F. Now let a secant is drawn from P to intersect the circle at G and H. PS is the tangent line from point P to S. Now, the formula for tangent and secant of the circle could be given as:

PH PS = PS PG

PS² = PH. PG

Read More: Circle Formulas

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Things to Remember

• At a single point tangent always touches the circle.
• At the point of tangency, it is perpendicular to the radius of the circle.
• A tangent can never cross a circle, which means that it cannot pass through a circle.
• At two points tangents never intersects the circle.
• The length of tangents from an external point to a circle are always equal.
• The line of the tangent is always perpendicular the radius of a circle.

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