MathematicsCirclesClass 10

Circles & tangent theorems (10.1 and 10.2): interactive simulation

A tangent to a circle is a straight line that touches the circle at exactly one point, known as the point of contact.

Useful for: CBSE Boards, ICSE Class 10

Try this

Theorem 10.1 states that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Theorem 10.2 proves that the lengths of tangents drawn from an external point to a circle are equal (PA = PB).

  1. 1

    Drag the distance slider (d) of point P and compare PA with PB.

    What you should see

    No matter the distance or orientation, PA is always identically equal to PB (Theorem 10.2)!

  2. 2

    Notice the angle between tangents (∠APB) and center angle (∠AOB).

    What you should see

    Their sum is always exactly 180° because the opposite angles at A and B are both 90° in quadrilateral OAPB.

Formulas behind it

Pythagorean Tangent Length

PA=PB=OP2−r2PA = PB = \sqrt{OP^2 - r^2}

Perpendicularity (Theorem 10.1)

∠OAP=∠OBP=90∘\angle OAP = \angle OBP = 90^\circ

Supplementary Angles Property

∠APB+∠AOB=180∘\angle APB + \angle AOB = 180^\circ

Common mistakes

  • Thinking a tangent can touch a circle at multiple points. A line intersecting a circle at two points is a Secant, not a Tangent.
  • Assuming tangents can be drawn from an interior point. From a point inside a circle, ZERO tangents can be drawn.
Stuck on a circles & tangent theorems (10.1 and 10.2) question?Snap it and get a whiteboard video solution.

FAQ

How many tangents can be drawn from a point outside a circle?

Exactly two tangents can be drawn from an external point to a circle.

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