Theory:

Learn how to construct a triangle with an example if its base, vertical angle and the point on the base where the bisector of the vertical angle meets the base are given.
Example:
Draw a triangle \(ABC\) of base \(AB = 7 \ cm\), \(\angle C = 30^{\circ}\) and the bisector of \(\angle C\) meets \(AB\) at \(C\) such that \(AD = 5 \ cm\).
 
Solution:
 
First, let us draw a rough figure.
 
11.PNG
 
Construction:
 
10.PNG
 
Step 1: Draw a line segment \(AB = 7 \ cm\).
 
Step 2: At \(A\), draw \(AE\) such that \(\angle EAB = 30^{\circ}\).
 
Step 3: At \(A\), draw \(AF\) such that \(\angle FAE = 90^{\circ}\).
 
Step 4: Draw the perpendicular bisector to \(AB\), which intersects \(AF\) at \(O\) and \(AB\) at \(P\).
 
Step 5: With \(O\) as centre and \(OA\) as radius, draw a circle.
 
Step 6: From \(A\), mark an arc of \(5 \ cm\) on \(AB\) at \(D\).
 
Step 7: The perpendicular bisector intersects the circle at \(R\). Join \(RD\).
 
Step 8: \(RD\) produced meets the circle at \(C\). Now, join \(AC\) and \(AB\).
 
Thus, \(\triangle ABC\) is the required triangle.