Question

The coordinates of base $BC$ of an isosceles triangles $ABC$ are given by $B(1,3)$ and $C(-2,7)$. Which of the following points can be the possible coordinates of the vertex $A$ ?

• A. $(-7,\frac{1}{8})$
• B. $(0,4,6)$
• C. $(-\frac{1}{2},5)$
• D. $(-\frac{5}{6},6)$

Answer 1

The correct option is B $(0,4,6)$

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 ............. (1)

And also vertex A lies on y-axis.

Let A be (0,y) = (x₁,y₁)

Equation (1) ⇒ AB = AC

√ [(x₂ - x₁)² + (y₂ - y₁)²] = √ [(x₂ - x₁)² + (y₂ - y₁)²]

(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

$y=\frac{41}{9}$

y = 4.6

So, co-ordinates of the vertex A are A (0,4.6)