Draw a circle of radius 4 cm and construct the pair of tangents to the circle from an external point, which is at a distance of 7 cm from its centre.

Answer

To solve the problem of drawing a circle of radius 4 cm and constructing the pair of tangents from an external point that is 7 cm away from the center of the circle, follow these steps:

### Step-by-Step Solution:

1. **Draw the Circle:**
- Using a compass, draw a circle with a radius of 4 cm. Label the center of the circle as point O.

2. **Mark the External Point:**
- From point O, measure a distance of 7 cm in any direction and mark this point as P. This point P is the external point from which the tangents will be drawn.

3. **Draw a Line Segment OP:**
- Draw a straight line segment connecting point O (the center of the circle) to point P (the external point).

4. **Find the Midpoint M of OP:**
- Measure the length of OP, which is 7 cm. The midpoint M can be found by measuring 3.5 cm from either O or P along the line segment OP. Mark this point M.

5. **Draw a Circle with Center M:**
- Using the compass, draw a circle with center M and radius equal to the length OM (which is 4 cm). This circle will intersect the line segment OP.

6. **Mark the Intersection Points:**
- Let the intersection points of the circle with the line segment OP be A and B. These points A and B are where the circle intersects the line OP.

7. **Draw Perpendicular Bisector:**
- Draw a perpendicular line to OP at point M. This line will help in finding the points where the tangents will touch the circle.

8. **Locate Points of Tangency:**
- The points where this perpendicular line intersects the circle will be the points of tangency. Label these points as T1 and T2.

9. **Draw the Tangents:**
- Finally, draw straight lines from point P to points T1 and T2. These lines PT1 and PT2 are the required tangents to the circle from the external point P.

### Final Construction:
- You should now have a circle with radius 4 cm, an external point P 7 cm away from the center O, and two tangents PT1 and PT2 touching the circle at points T1 and T2.