6. Draw a circle of radius 2.7 cm and draw chord PQ of length 4.5 cm. Draw tangents at P and Q without using centre.

Q6. Draw a circle of radius 2.7 cm and draw chord PQ of length 4.5 cm. Draw tangents at P and Q without using the centre.

🎓 Concept: Tangent-Secant Theorem (Alternate Segment Theorem)

To draw tangents without using the centre, we use the property that the angle between a tangent and a chord is equal to the angle in the alternate segment.

If we construct a triangle \( \Delta PQR \) inside the circle, the angle required for the tangent at point P will be equal to \( \angle R \), and similarly for point Q.

Steps of Construction:

1. Draw a circle with a radius of 2.7 cm.
2. Take a point P anywhere on the circle.
3. Using a compass, take a distance of 4.5 cm, place the metal point on P, and cut an arc on the circle to mark point Q. Join chord PQ.
4. Take any point R on the major arc (the larger side of the circle) and join PR and QR to form \( \Delta PQR \).
5. Place the compass at point R and draw an arc intersecting sides PR and QR. Keep the same radius.
6. Place the compass point at P and draw a similar arc intersecting chord PQ. Do the same at point Q intersecting chord QP.
7. Measure the distance of the arc drawn at angle R. Cut this distance on the arcs drawn at P and Q to replicate \( \angle PRQ \).
8. Draw a line passing through P and the intersection point of the arcs. This is the required tangent at P.
9. Draw a line passing through Q and the intersection point of the arcs. This is the required tangent at Q.

Visual Guide:

Fig 1: Rough Figure (Planning the Construction)

Fig 2: Final Construction (Fair Figure)