### Theory:

Let us learn how to construct a triangle with an example when its base, vertical angle and the median from the vertex of the base are given.
Example:
Construct a triangle $$ABC$$ in which $$AB = 9.6 \ cm$$, $$\angle C = 48^{\circ}$$ and the median $$CQ$$ from $$C$$ to $$AB$$ is $$9.6 \ cm$$. Find the length of the altitude from $$C$$ to $$AB$$.

Solution:

First, let us draw a rough figure.

Construction:

Step 1: Draw a line segment $$AB$$ of length $$9.6 \ cm$$.

Step 2: At $$A$$, draw $$AD$$ such that $$\angle DAB = 48^{\circ}$$.

Step 3: At $$A$$, draw $$AE$$ such that $$\angle DAE = 90^{\circ}$$.

Step 4: Draw the perpendicular bisector of $$AB$$, which intersects $$AE$$ at $$P$$ and $$AB$$ at $$Q$$.

Step 5: With $$P$$ as centre and $$AP$$ as radius, draw a circle.

Step 6: From $$Q$$, mark arcs of radius $$9.6 \ cm$$ on the circle. Mark them as $$C$$ and $$R$$.

Step 7: Join $$AC$$ and $$BC$$. Thus, $$\triangle ABC$$ is the required triangle.

Step 8: From $$C$$, draw a line $$CW$$ perpendicular to $$BV$$. $$BV$$ meets $$CW$$ at $$Z$$.

Step 9: The length of the altitude is $$CZ = 8 \ cm$$.