Find the value of k, if the points (5, k) and (k, 7) are equidistant from point (2, 4).

Given (2, 4) is equidistant from (5, k) and (k, 7).

i.e. distance between (2, 4) and (5, k) = distance between (2, 4) and (k, 7)

$\sqrt{(2 - 5)^2 + (4 - k)^2} = \sqrt{(2 - k)^2 + (4 - 7)^2}\\[1em] ⇒ (2 - 5)^2 + (4 - k)^2 = (2 - k)^2 + (4 - 7)^2\\[1em] ⇒ (- 3)^2 + (4 - k)^2 = (2 - k)^2 + (- 3)^2\\[1em] ⇒ 9 + 16 + k^2 - 8k = 4 + k^2 - 4k + 9\\[1em] ⇒ 25 + k^2 - 8k = 13 + k^2 - 4k\\[1em] ⇒ 13 + k^2 - 4k - 25 - k^2 + 8k = 0\\[1em] ⇒ - 12 + 4k = 0\\[1em] ⇒ 4k = 12\\[1em] ⇒ k = \dfrac{12}{4}\\[1em] ⇒ k = 3$

Hence, the value of k = 3.