Concept 10.3
Tangents from an Outside Point Are Equal
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
Theorem 10.2: the lengths of tangents drawn from an external point to a circle are equal.
The intuition
From an outside point P, two tangents touch the circle at A and B. Triangles OAP and OBP are right-angled at A and B, share the side OP, and have OA = OB.
So they are congruent, and PA = PB. Tangent lengths from one point always come in equal pairs, the fact most tangent problems turn on.
The attention
Look at the two tangents from the same point: they are equal.
PA = PB not PA = PO
Look hereNot here
- Look here
- The two tangent lengths from one point. PA and PB are equal, so one known length gives you the other for free.
- Not here
- The line to the centre. PO goes to the centre and is longer than either tangent. It is the hypotenuse of both right triangles.
- What it teaches
- Symmetry gives information for free. Look for the mirror line.