If the points A(3, 0), B(x, 5), C(-1, 4) and D(-2, -1) are the vertices of a rhombus, taken in order, find the value of x.

The diagonals of a rhombus bisect each other at the same point.

The vertices are in order: A → B → C → D

The diagonals are:

• Diagonal 1: A to C
• Diagonal 2: B to D

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Given: A(3, 0) and C(-1, 4)

Midpoint of AC:

$= \left(\dfrac{3 + (-1)}{2}, \dfrac{0 + 4}{2}\right)$

$= \left(\dfrac{2}{2}, \dfrac{4}{2}\right)$

$= (1, 2)$

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Given: B(x, 5) and D(-2, -1)

Midpoint of BD:

$= \left(\dfrac{x + (-2)}{2}, \dfrac{5 + (-1)}{2}\right)$

$= \left(\dfrac{x - 2}{2}, \dfrac{4}{2}\right)$

$= \left(\dfrac{x - 2}{2}, 2\right)$

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Since both diagonals bisect at the same point:

$(1, 2) = \left(\dfrac{x - 2}{2}, 2\right)$

The y-coordinates match.

For x-coordinates:

$1 = \dfrac{x - 2}{2}$

$2 = x - 2$

$x = 4$

Therefore, the value of $x = 4$.