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.

Two Tangents from an External Point

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.

Tangents from an External Point

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.





10th Grade Math

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