In a rhombus ABCD, O is any interior point such that OA = OC. What is five-ninths of the degree measure of ∠ DOB?

Correct — B. The condition OA = OC puts O on the perpendicular bisector of diagonal AC.

In a rhombus the diagonals bisect each other at right angles, so BD is itself the perpendicular bisector of AC.

O must therefore lie on BD, which makes D, O and B collinear.
∠DOB = 180° (a straight angle)
5⁄9 × 180° = 100° → option (b)

Why the others are wrong

• (a)
  150° — 150° is five-sixths of 180°, not five-ninths. The geometry has been read correctly and the fraction misapplied.
• (c)
  90° — 90° is the angle at which the diagonals of a rhombus cross — a remembered property handed back instead of used. The question asks for a fraction of ∠DOB, not for the crossing angle.
• (d)
  120° — 120° is two-thirds of 180°. Multiplying the straight angle by 2⁄3 instead of 5⁄9 is the one slip that survives an otherwise complete argument.

The whole question turns on one locus fact: the points equidistant from A and C are exactly the points of the perpendicular bisector of AC.

A rhombus has four equal sides, so both B and D are equidistant from A and C, and the diagonal BD is that perpendicular bisector.

Saying OA = OC therefore says nothing about where O sits *along* BD — only that it sits on it. Any interior point of BD gives the same straight angle, which is why the answer is a single number and not a range.

Key facts

• In a rhombus the diagonals bisect each other at 90°, so each diagonal is the perpendicular bisector of the other.
• The set of points equidistant from two fixed points is the perpendicular bisector of the segment joining them.
• If O lies between D and B on a straight line, ∠DOB is the straight angle 180°.
• Five-ninths of 180° is 100°.

Common traps

• Answering the crossing angle 90° and never applying the five-ninths the stem asks for
• Assuming an arbitrary interior point makes ∠DOB indeterminate, and skipping the question

No directly related past PYQ was found.