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.
From Two Tangents from an External Point to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Oct 08, 24 10:53 AM
Oct 07, 24 04:07 PM
Oct 07, 24 03:29 PM
Oct 07, 24 03:13 PM
Oct 07, 24 12:01 PM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.