Solution

From the given question,
\[AC=6\]cm and \[BC=8\] cm
We know that a triangle in a semi-circle with hypotenuse as diameter is right angled triangle.
By using Pythagoras theorem in triangle ACB,
\[{{(AB)}^{2}}={{(AC)}^{2}}+{{(CB)}^{2}}\]
= \[{{(AB)}^{2}}={{(6)}^{2}}+{{(8)}^{2}}\]
⇒\[{{(AB)}^{2}}=36+64\]
⇒\[{{(AB)}^{2}}=100\] ⇒\[{{(AB)}^{2}}=100\]
Therefore, Diameter of the circle = \[10\] cm
from above diameter we get Radius of the circle = \[5\] cm
Area of circle = \[\pi {{r}^{2}}\]
= \[\pi {{(5)}^{2}}\]
= \[25\pi \] \[c{{m}^{2}}\]
= \[25\times 3.14\] \[c{{m}^{2}}\]
= \[78.5\] \[c{{m}^{2}}\]
We know that,
Area of the right angled triangle = \[(\frac{1}{2})\times Base\times Height\]
= \[(\frac{1}{2})\times AC\times CB\]
= \[(\frac{1}{2})\times 6\times 8\]
= \[24\] \[c{{m}^{2}}\]
Now Area of the shaded region can be determined by (Area of the circle – Area of the triangle)
= \[(78.5-24)\] \[c{{m}^{2}}\]
= \[54.5\] \[c{{m}^{2}}\]