Geometry

Quadrilaterals

Medium

If the points P(8,5), Q(4,8), R(0,5) and S(4,k) are the vertices of a rhombus, taken in order, then the value of k is

✅ Correct Option: 4

For a rhombus, all four sides are equal in length and the diagonals bisect each other at their midpoint.

Since the vertices are given in order P→Q→R→S, the diagonals are PR and QS.

The diagonals must bisect each other, so:

Midpoint of PR = Midpoint of QS

Finding the midpoint of diagonal PR:

P = (8, 5) and R = (0, 5)

Midpoint of PR $= \left(\dfrac{8+0}{2}, \dfrac{5+5}{2}\right)$

$= (4, 5)$

Finding the midpoint of diagonal QS:

Q = (4, 8) and S = (4, k)

Midpoint of QS $= \left(\dfrac{4+4}{2}, \dfrac{8+k}{2}\right)$

$= \left(4, \dfrac{8+k}{2}\right)$

Setting the midpoints equal:

$(4, 5) = \left(4, \dfrac{8+k}{2}\right)$

The x-coordinates match. Equating the y-coordinates:

$5 = \dfrac{8+k}{2}$

$10 = 8 + k$

$k = 2$

Therefore, the value of $k = 2$ .

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If the points P(8,5), Q(4,8), R(0,5) and S(4,k) are the vertices of a rhombus, taken in order, then the value of k is

Solution

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