First slide

Cartesian plane

Question

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

Moderate

A

(3, 1)
B

(3, 2)
C

(3, 4)
D

(3, 6)

Solution

Let the coordinates of the other vertices lying on $x = 3$ be $(3, y_{1}) and (3, y_{2})$
Since the diagonals of a rectangle are equal and bisect each other, we have
$\frac{y_{1} + y_{2}}{2} = \frac{5 + 2}{2} \Rightarrow y_{1} + y_{2} = 7$
and $|y_{2} - y_{1}| = \sqrt{(5 - 1)^{2} + (5 - 2)^{2}} = \sqrt{16 + 9} = 5$
If $y_{2} > y_{1} then y_{2} = 6, y_{1} = 1$
So that the coordinates of the vertex nearer the axis of $x$ are (3, 1)

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