Question

$ABC$ is an isosceles triangle. If the coordinates of the base are B $(1,3)$ and $C(-2,7)$ then the vertex A can be

• A. $(4,6)$
• B. $(-1/2,5)$
• C. $(-7,1/6)$
• D. none of these

Answer 1

Answer: (D)

none of these

Given that $ABC$ is an isosceles triangle.

The two vertices are $B(1,3)=(x₂,y₂)$ and $C(-2,7)=(x₂,y₂)$

We have to find the co-ordinates of the vertex$A$

Since the triangle is an isosceles triangle So,

$AB=AC.............\left(1\right)$

And also vertex $A$ lies on y-axis.

Let $A$ be $(0,y)=(x₁,y₁)$

Equation (1) $\Rightarrow AB=AC$

$\surd \left[\right(x₂-x₁)²+(y₂-y₁\left)²\right]=\surd \left[\right(x₂-x₁)²+(y₂-y₁\left)²\right]$

$(1-0)²+(3-y)²=(-2-0)²+(7-y)²$

$1+(3-y)²=2+(7-y)²$

$1+9+y²-6y=2+49+y²-14y$

$10-6y=51-14y$

$14y-6y=51-10$

$9y=41$

$y=41/9$

$y=4.6$

So, co-ordinates of the vertex A are $A(,4.6)$

Hence, the option $A$ is the correct answer.