Respuesta :
Answer: [tex](-5.5,4.062)\ \text{and}\ (-5.5,-8.062)[/tex]
Step-by-step explanation:
Given
The two vertices of a equilateral triangle are [tex](-9,-2)\ \text{and}\ (-2,-2)[/tex]
Suppose the third vertex is [tex](x,y)[/tex]
Side length in equilateral triangle are equal.
[tex]\Rightarrow \sqrt{\left( -2+9\right)^2+\left( -2+2\right)^2}=\sqrt{\left( x+9\right)^2+\left( y+2\right)^2}\\\text{squaring both sides}\\\\\Rightarrow 7^2=\left( x+9\right)^2+\left( y+2\right)^2\quad \ldots(i)\\\\[/tex]
Similarly,
[tex]\Rightarrow \sqrt{\left( -2+9\right)^2+\left( -2+2\right)^2}=\sqrt{\left( x+2\right)^2+\left( y+2\right)^2}\\\\\Rightarrow 7^2=\left( x+2\right)^2+\left( y+2\right)^2\quad \ldots(ii)[/tex]
Subtract (i) and (ii)
[tex]\Rightarrow \left( x+9\right)^2-\left(x+2\right)^2=7^2-7^2\\\\\Rightarrow (x+9+x+2)(x+9-x-2)=0\\\Rightarrow (2x+11)7=0\\\\\Rightarrow x=-\dfrac{11}{2}[/tex]
Insert the value of [tex]x[/tex] in equation (ii)
[tex]\Rightarrow \left(-5.5+2\right)^2+\left(y+2\right)^2=7^2\\\\\Rightarrow \left(-5.5+2\right)^2+\left(y+2\right)^2-\left(-5.5+2\right)^2=7^2-\left(-5.5+2\right)^2\\\\\Rightarrow \left(y+2\right)^2=36.75\\\\\Rightarrow y=4.06217\dots ,\:y=-8.06217[/tex]
So, two possible vertex are [tex](-5.5,4.062)\ \text{and}\ (-5.5,-8.062)[/tex]
