C
CraftExam
medium difficulty5 min

Equal tangents from an external point

Tangents drawn from the same external point to a circle are equal in length.

In one line: Same outside point, same tangent length.

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Understand it

If an external point P touches the circle at A and B, then PA = PB. This theorem is extremely important in proofs and length-finding questions. It is not enough to know that the tangents exist; you must also remember that their lengths are equal.

Example

From external point P, if PA and PB are tangents to the circle, then PA = PB.

Formula and rule

If PA and PB are tangents from the same external point P, then PA = PB.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

If PA = 7 cm and PB is another tangent from the same outside point P, then PB is also 7 cm.

Method you can reuse

  1. 1Identify the external point.
  2. 2Check that both lines are tangents.
  3. 3Use the theorem that tangents from the same external point are equal.
  4. 4Substitute the equal lengths and solve.

Sketch it in your answer

Draw the outside point, the two tangent segments, and label both contact points clearly.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Two tangents are drawn from the same point outside a circle. What can be said about their lengths?

Think first, then show answer
Their lengths are equal. Tangents drawn from the same external point to a circle are always equal in length.

Common wrong answer

A common wrong answer is to compare tangents from different points or to say only the arcs are equal. The theorem is about tangent lengths from the same outside point.

Don't mix this up

Students sometimes treat tangent lengths from different external points as equal. The theorem works only for tangents from the same external point.

Exam tip

Whenever two tangents start from one outside point, immediately write their lengths equal.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

This theorem is often asked in proof steps and in finding an unknown tangent length.
  1. 1

    Enough for 1 mark

    Tangents drawn from the same external point to a circle are equal in length.

  2. 2

    Add for 2 marks

    Example: From external point P, if PA and PB are tangents to the circle, then PA = PB.

  3. 3

    Add for 3 marks

    If PA = 7 cm and PB is another tangent from the same outside point P, then PB is also 7 cm.

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