Given points(-1,1),and (-11,-9)form the vertices of the base of a triangle,find a third point so that the three points form the vertices of an isosceles triangle.
A)(3,1)
B)(4,-15)
C)(-15,-13)
D)(3,5)
E)(-6,-4)

Question

Given points(-1,1),and (-11,-9)form the vertices of the base of a triangle,find a third point so that the three points form the vertices of an isosceles triangle. ​
A)(3,1)
B)(4,-15)
C)​(-15,-13)
D)​(3,5)
E)​(-6,-4)

Quizplus · Accepted Answer

First, we find the midpoint of the line segment connecting the two given points:
Midpoint = $\left(\frac{-1-11}{2},\frac{1-9}{2}\right) = (-6,-4)$
Then, we need to find a point on the same line passing through the given two points, which is equidistant from both the given points. The distance between the given points is $\sqrt{(11-(-1))^2+((-9)-1)^2}=\sqrt{144+100}=2\sqrt{61}$.
Let the third point be $(x,y)$. Since it is equidistant from the given points, the distance from $(x,y)$ to $(-1,1)$ and from $(x,y)$ to $(-11,-9)$ must be equal:
$\sqrt{(x-(-1))^2+(y-1)^2}=\sqrt{(x-(-11))^2+(y+9)^2}$
Simplifying this equation, we get:
$x+y=2$
Solving the above equation and the equation of the line passing through the given two points: $y=x-2$, we get the coordinates of the third point as $(-6,-4)$ and $(3,5)$. Since $(3,5)$ is not given as a choice, the answer is (E).

Given Points(-1,1),and (-11,-9)form the Vertices of the Base of a Triangle,find