Question

Construct a transverse tangent on two equal circle A & B of radius 3 cm each and distance between their centres is 7 cm.

Hint:

For a transverse tangent, the key step is creating a construction circle with a radius equal to the SUM of the two given radii (r\textsubscript{A} + r\textsubscript{B}). For a direct tangent, the radius would be the DIFFERENCE (r\textsubscript{A} - r\textsubscript{B}).

Answer 1

Step 1: Understanding the Task
The task is to construct a transverse (or internal) common tangent to two circles of equal radius. A transverse tangent crosses the line segment connecting the centers of the two circles.
Step 2: Steps of Construction
Draw a horizontal line. On this line, mark two points A and B such that the distance between them is 7 cm. These are the centers of the circles.
With A and B as centers, draw the two given circles, each with a radius of 3 cm.
Construct the perpendicular bisector of the line segment AB to find its midpoint. Let's call the midpoint M.
With M as the center and MA (or MB) as the radius, draw a semicircle (or a full circle) on the line segment AB.
Now, we create a construction circle. With A as the center, draw a circle with a radius equal to the sum of the radii of the two given circles (r\textsubscript{A} + r\textsubscript{B}). So, the radius of this construction circle is 3 cm + 3 cm = 6 cm.
This large construction circle will intersect the semicircle drawn in step 4 at a point. Let's label this point P.
Draw a straight line from center A through point P. This line will intersect the original circle (centered at A) at a point. Label this point T\textsubscript{1}. This is the point of tangency on the first circle.
Now, from center B, draw a line parallel to the line AP but in the opposite direction (downwards if AP went upwards). This parallel line will intersect the original circle (centered at B) at a point. Label this point T\textsubscript{2}. This is the point of tangency on the second circle. (An easy way to draw this is to use a set square and ruler).
Join the points T\textsubscript{1} and T\textsubscript{2} with a straight line.
The line segment T\textsubscript{1T\textsubscript{2}} is the required transverse common tangent.