Question

If the point A (7, k) is the vertex of an isosceles triangle ABC with base BC,
where B = (2, 4) and C = (6, 10), then what is 'k'?
1) 6 2) 3 3) 4 4) 5

Answer 1

A , B and C are vertices of an isosceles triangle with base BC.
so, AB = BC
use distance formula,
AB = $\bf{\sqrt{(7-2)^2+(k-4)^2}}$
AB = $\bf{\sqrt{5^2+(k-4)^2}}$

similarly, BC = $\bf{\sqrt{(7-6)^2+(k-10)^2}}$
BC = $\bf{\sqrt{1+(k-10)^2}}$

so, $\bf{\sqrt{5^2+(k-4)^2}=\sqrt{1+(k-10)^2}}$
squaring both sides,
5² + (k - 4)² = 1² + (k - 10)²
25 + (k - 4)² = 1 + (k - 10)²
24 = (k - 10)² - (k - 4)²
24 = -6(2k - 14)
-4 = 2k - 14
10 = 2k
k = 5

hence, option (4) is correct