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  • State whether the statements are true or false. The vertex of an equilateral triangle is (2, 3) and the equation of the opposite side is x + y = 2. Then the other two sides are y-3=(2 +- √3)(x-2).

State whether the statements are true or false.

The vertex of an equilateral triangle is (2, 3) and the equation of the opposite side is x + y = 2. Then the other two sides are y-3=\left ( 2\pm \sqrt{3} \right )\left ( x-2 \right ).

Answers (1)

Let ABC be an equilateral triangle with vertex (2,3) and the equation of the opposite side is x+y=2  

In the case of equilateral triangle θ=60  

 Let the slope of line AB is m and the slope  of the given equation x+y=2 is m2=-1

 We know that \tan \theta =\left | \frac{\left ( m_{1}-m_{2} \right )}{1+m_{1}m_{2}} \right |

  Putting the values of m1 and m2 in the above equation, we get \tan 60^{\circ}=\left | \frac{\left ( m-\left ( -1 \right ) \right )}{1+m\left ( -1 \right )} \right |  

\sqrt{3}=\left | \frac{m+1}{1-m}\right | 

\sqrt{3}=\pm \left (\frac{m+1}{1-m} \right )

   \sqrt{3}= \left (\frac{m+1}{1-m} \right ) 0r -\left (\frac{m+1}{1-m} \right )

 \left ( 1-m \right )\sqrt{3}=1+m \, \, or \, \, \left ( 1-m \right )\left (-\sqrt{3} \right )=-1-m 

\sqrt{3}-\sqrt{3}m=1+m \, \, or\, \, -\sqrt{3}+\sqrt{3}m=-1-m

  \sqrt{3}-1=m \left ( 1+\sqrt{3} \right )\, \, or\, \, -\left (\sqrt{3}-1 \right )=-\left ( \sqrt{3} +1\right )   

m=\frac{\sqrt{3}+1}{\sqrt{3}-1}...........(i)\, \, \, or\, \, m=\frac{\sqrt{3}-1}{\sqrt{3}+1}...........(ii)

 Rationalizing both the equations we getm=2+ \sqrt{3} \, \, \, or\, \, m=2- \sqrt{3}

 So, the slope of line AB is 2\pm + \sqrt{3}

  Thus, the equations of the other two lines joining the point (2, 3) are  y-3=2\pm \sqrt{3}\left ( x-2 \right )

  Hence, the given statement is True. 

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infoexpert22

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