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Question

If the vertices of a triangle are A(2,4),B(2,6) and C(2+3,5), then which of the following is/are true?

A
Coordinates of the excentre opposite to vertex A is (2+3,7)
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B
Coordinates of the excentre opposite to vertex B is (23,3)
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C
Coordinates of the excentre opposite to vertex C is (23,5)
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D
The area of ABC is 3 sq. units
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Solution

The correct option is D The area of ABC is 3 sq. units
Given vertices of the triangle are A(2,4),B(2,6) and C(2+3,5)
Now, finding the side lengths of the triangle,
a=BC=(3)2+12=2 units
b=AC=(3)2+12=2 units
c=AB=02+22=2 units

So, ABC is an equilateral.
Now, we know that
Excentre opposite to vertex A
=(ax1+bx2+cx3a+b+c,ay1+by2+cy3a+b+c)
=(x1+x2+x3,y1+y2+y3)
=(2+3,7)

Excentre opposite to vertex B
=(ax1bx2+cx3ab+c,ay1by2+cy3ab+c)
=(x1x2+x3,y1y2+y3)
=(2+3,3)

Excentre opposite to vertex C
=(ax1+bx2cx3a+bc,ay1+by2cy3a+bc)
=(x1+x2x3,y1+y2y3)
=(23,5)

Now, area of an equilateral triangle
=3a24=3×224=3 sq. units

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