Question

ABC is an equilateral triangle with coordinates of B and C as B(-4,0) and C(4,0). Find the coordinate of vertex A.

Answer 1

clearly A lies on y-axis. because mid point of BC is origin.

Now length of BC = 8 = AB = AC

So AO =

\sqrt{ {8}^{2} - {4}^{2} } = \sqrt{64 - 16} = \sqrt{48} = 4 \sqrt{3}

8

2

−4

2

=

64−16

=

48

=4

3

so coordinate of A is (0 , 4√3) or (0 , -4√3)

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Answer 2

Let A(x,y), B(-4,0), C(4,0)

Distance between BC = 8 (Calculate it)

Distance between AB = √ (-4-x)^2 + (0-y)^2

Distance between AC = √ (4-x)^2 + (0-y)^2

Given that the triangle is equilateral. So, AB=BC= AC

AB=AC

√(-4-x)^2+ (0-y)^2 = √ (4-x)^2+ (0-y)^2

(-4-x)^2+ (0-y)^2 = (4-x)^2+ (0-y)^2

(-4-x)^2= (4-x)^2

x^2+8x+16= x^2-8x+16

8x+8x= 16-16

x= 0 ..........(1)

Again, AC = BC

√(4-x)^2+ (0-y)^2 = 8

(4-x)^2+ (0-y)^2= 64

(4-0)^2+ (0-y)^2 { substituting 1}= 64

4^2 + y^2 = 64

y^2 = 64-16=38

y = +√38 or -√38

Therefore, x=0 and y = +√38 or -√38

Third vertex is (0, √38) or (0,-√38)