Construct ∆ABC, AB=6cm, BC=9cm,CA=8cm. construct tangents from the vertex C to the circle through AB as the diameter

Assuming you can construct the triangle, the excellent web site at

http://www.mathopenref.com/consttangents.html

will show how to construct the tangents to the circle.

To construct the tangents from the vertex C to the circle through AB as the diameter, follow these steps:

Step 1: Draw a line segment AB of length 6 cm.
Step 2: From point A, draw a circle with a radius of 3 cm (half the length of AB), and label the center of the circle as O.
Step 3: From point B, draw another circle with a radius of 4.5 cm (half the length of BC), and label the center of this circle as P.
Step 4: Extend the line segment BA beyond point A and label the extended line as l.
Step 5: Find the point D, which is the intersection of the line l and the circle centered at point O.
Step 6: Draw a line segment CD.
Step 7: From point C, draw a line perpendicular to CD, intersecting the circle centered at point P. Label this point of intersection as E.
Step 8: Draw a line segment CE.
Step 9: The line segment CE and the line segment CD are the tangents from vertex C to the circle with AB as the diameter.

By following these construction steps, you will have successfully constructed the tangents from the vertex C to the circle with AB as the diameter.