Construction: tangents to a circle

Steps of construction: tangent at a point, using the centre

Question: draw a circle of radius 3.5 cm. Take a point P on it and draw the tangent at P. Instruments: ruler, compasses, sharp pencil.

1. 1
   Draw a rough figure. Show a circle with centre O, a point P on it, the radius OP, and the tangent TT′ at right angles to OP.
2. 2
   Mark O. With O as centre and radius 3.5 cm, draw the circle.
3. 3
   Take any point P on the circle. Join OP and extend it beyond P.
4. 4
   With P as centre and a small radius, cut the line OP at two points, one on each side of P.
5. 5
   With these two points as centres and a larger radius, draw arcs that cut each other at Q.
6. 6
   Join PQ and extend it both ways to T and T′. Now TT′ ⊥ OP.
7. 7
   TT′ is the required tangent at P.

Steps of construction: tangent at a point, using the alternate segment theorem

Question: draw a circle of radius 3 cm. At a point P on it, draw the tangent without using the centre. Instruments: ruler, compasses, sharp pencil.

1. 1
   Draw a rough figure. Show a circle, a point P, a chord PQ, a point R on the major arc, and the tangent TT′ at P.
2. 2
   Draw a circle of radius 3 cm. Take a point P on it.
3. 3
   Draw any chord PQ through P.
4. 4
   Take a point R on the major arc. Join PR and RQ.
5. 5
   With R as centre and any radius, draw an arc cutting RP at L and RQ at M.
6. 6
   With P as centre and the same radius, draw an arc on the side of PQ away from R. It cuts PQ at N.
7. 7
   With N as centre and radius LM, cut this arc at S. Join PS and extend it both ways to T and T′.
8. 8
   TT′ is the required tangent, because ∠QPT = ∠PRQ.

   alternate segment theorem

Steps of construction: pair of tangents from an outside point

Question: draw a circle of radius 4 cm. From a point P, 9 cm from the centre, draw the two tangents and measure them. Instruments: ruler, compasses, sharp pencil.

1. 1
   Draw a rough figure. Show a circle with centre O, a point P outside it, the midpoint M of OP, and the tangents PA and PB.
2. 2
   With centre O and radius 4 cm, draw the circle.
3. 3
   Mark P with OP = 9 cm. Join OP.
4. 4
   Draw the perpendicular bisector of OP. It cuts OP at M, the midpoint.
5. 5
   With M as centre and radius MO (4.5 cm), draw a circle. It cuts the first circle at A and B.
6. 6
   Join PA and PB. These are the required tangents.
7. 7
   Measure them: PA = PB ≈ 8.1 cm.

Example: verify the tangent length

Why it works

In the third construction, A lies on the circle with diameter OP. So ∠OAP = 90° (angle in a semicircle). PA is perpendicular to the radius OA at A, so PA is a tangent. The same holds for PB. In the second construction, the copied angle equals the angle in the alternate segment. So, by the alternate segment theorem used in reverse, TT′ must be a tangent.