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Area of the largest triangle that can be inscribed in a semi-circle of radius r units is (A) \[{{r}^{2}}\] sq. units (B) \[\frac{1}{2}{{r}^{2}}\] sq. units (C) \[2{{r}^{2}}\] sq. units (D) \[\sqrt{2}{{r}^{2}}\] sq. units
Area of the largest triangle that can be inscribed in a semi-circle of radius r units is (A) \[{{r}^{2}}\] sq. units (B) \[\frac{1}{2}{{r}^{2}}\] sq. units (C) \[2{{r}^{2}}\] sq. units (D) \[\sqrt{2}{{r}^{2}}\] sq. units
Areas Related to Circles | Exercise 11.1 | Maths | NCERT Exemplar
Area of the largest triangle that can be inscribed in a semi-circle of radius r units is (A) \[{{r}^{2}}\] sq. units (B) \[\frac{1}{2}{{r}^{2}}\] sq. units (C) \[2{{r}^{2}}\] sq. units (D) \[\sqrt{2}{{r}^{2}}\] sq. units

The correct option is (A) \[{{r}^{2}}\] 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 , \[\angle C={{90}^{\circ }}\] (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 \[\vartriangle ABC=(1/2)\times Base\times Height=(1/2)\times AB\times CD\]

We got \[(1/2)\times 2r\times r\]

= \[{{r}^{2}}\] sq. units

Hence Option A is correct.

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