Question

Draw a circle with radius 4.1 cm. Construct tangents to the circle from a point at a distance 7.3 cm from the centre.

Hint:

The key to this construction is finding the perpendicular bisector of the segment joining the center and the external point. The intersection of the new arc (or circle) with the original circle gives you the exact points of tangency.

Answer 1

Step 1: Understanding the Concept: 
This is a geometric construction problem. We need to construct tangents to a given circle from a point outside the circle using a compass and a straightedge. The construction relies on the property that the angle in a semicircle is a right angle, which ensures the tangent is perpendicular to the radius at the point of contact. 
 

Step 2: Key Formula or Approach: 
The steps for construction are as follows: 
Steps of Construction: \[\begin{array}{rl} 1. & \text{Draw a circle with centre O and radius 4.1 cm.} \\ 2. & \text{Take a point P in the exterior of the circle such that the distance OP = 7.3 cm.} \\ 3. & \text{Draw the line segment OP.} \\ 4. & \text{Construct the perpendicular bisector of the segment OP. Let M be the midpoint of OP.} \\ 5. & \text{With M as the centre and radius equal to MO (or MP), draw an arc that intersects the given circle at two distinct points, say A and B.} \\ 6. & \text{Draw the lines PA and PB.} \\ 7. & \text{Lines PA and PB are the required tangents to the circle from point P.} \\ \end{array}\] Justification (Optional but good to know): 
Join OA. \(\triangle\)OAP lies in the semicircle with diameter OP. Therefore, \(\angle\)OAP = 90\(^\circ\) (angle in a semicircle). Since OA is a radius and \(\angle\)OAP is 90\(^\circ\), the line PA must be a tangent to the circle at point A. Similarly, PB is a tangent at point B. 
 

Step 4: Final Answer: 
The construction should be performed on paper following the steps above to get the required tangents.