# Two Tangents from an External Point

Here we will prove that from any point outside a circle two tangents can be drawn to it and they are equal in length.

Given: O is the centre of a circle and T is a point outside the circle.

Construction: Join O and T. Draw a circle with TO as diameter which cuts the given circle at M and N. Join T to M and N.

To prove: TM and TN are tangent to the circle and TM = TN.

Proof:

 Statement Reason 1. ∠TMO = 90°. 1. Angle in a semicircle is a right angle. 2. TM ⊥ OM. 2. From statement 1. 3. Therefore, TM is a tangent to the given circle. 3. Tangent ⊥ radius drawn through point of contact. 4. Similarly, TN is a tangent to the given circle. 4. Proceeding as above. 5. In ∆TOM and ∆TON,(i) OM = ON.(ii) ∠OMT = ∠ONT = 90°.(iii) TO = TO. 5. (i) Radii of the same circle.(ii) Radius ⊥ tangent.(iii) Common side. 6. ∆TOM ≅ ∆TON. 6. By RHS criterion. 7. TM = TN. 7. CPCTC.

Note:

1. The two tangents subtend equal angles at the centre of the circle.

∠TOM = ∠TON, as ∆TOM ≅ ∆TON.

2. The two tangents are equally inclined to the line joining the point to the centre of the circle.

∠MTO = ∠NTO, as ∆TOM ≅ ∆TON.

Alternate Segments

In the given below figure, the chord MN divides the circle into two segments. The tangent XY is drawn that touches the circle N.

The alternate segment for ∠MNY is the segment MAN and that for ∠MNX is the segment MBN.

The angle in the alternate segment for ∠MNY is ∠MAN and that for ∠MNX is ∠MBN.