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Let us learn how to construct a triangle with an example when its base, vertical angle and 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**: Draw a circle with \(P\) as centre and \(AP\) as radius.

**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 \(CZ = 8 \ cm\).