![\[\frac{1}{2}{{r}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1096aebfec9f6ab10a80a57d78612071_l3.png "Rendered by QuickLaTeX.com")
![\[2{{r}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-75770605e69eb7b76ee0089f20723c17_l3.png "Rendered by QuickLaTeX.com")
![\[\sqrt{2}{{r}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4c00e2129d9d29553fc9048879bbd405_l3.png "Rendered by QuickLaTeX.com")
The correct option is (A)
![\[{{r}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d8ad9fba333e9a155e4460a7f414ad25_l3.png "Rendered by QuickLaTeX.com")
sq. units
Explanation:
The largest triangle which can be inscribed in a semi-circle of radius r units is
Base of triangle should be equal to the diameter of the semi-circle
The two other sides of triangle are taken by considering the point C on the circumference of the semi-circle and joining it by the end points of diameter A and B.
Therefore ,
![\[\angle C={{90}^{\circ }}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-52bdad80430b319981bc01013d6fa6e5_l3.png "Rendered by QuickLaTeX.com")
(by the properties of circle)
So, Triangle ABC is right angled triangle where base as diameter AB of the circle and height be CD.
Let Height of the triangle = r
Therefore, Area of largest
![\[\vartriangle ABC=(1/2)\times Base\times Height=(1/2)\times AB\times CD\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0dcbef2b41a9a78ce94437c2cee8e4d8_l3.png "Rendered by QuickLaTeX.com")
We got
![\[(1/2)\times 2r\times r\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-451bb2a89d193667142bc8a8d68454ba_l3.png "Rendered by QuickLaTeX.com")
=
sq. units
Hence Option A is correct.