A rectangle has two opposite vertices at the points (1, 2) and (5, 5). If the other vertices lie on the line x = 3, then the coordinates of the other vertices are

Let A ≡ (1, 2) and C ≡ (5, 5). Since the vertices B and D lie on the line x = 3, therefore, let B ≡ (3, y₁) and D ≡ (3, y₂).
Since AC and BD bisect each other, so they have same middle point

$i.e, \frac{y_1+y_2}{2}=\frac{2+5}{2}$

$or y_1+y_2=7$                                 (1)

Also, BD² = AC²

⇒ (y₁ – y₂)² = (1 – 5)² + (2 – 5)² = 25
or y₁ – y₂ = ± 5                                   (2)

Solving (1) and (2), we get y₁ = 6, y₂ = 1 or y₁ = 1, y₂ = 6
Thus, the other vertices of the rectangle are (3, 1)
and (3, 6).