Question
In a rhombus ABCD, O is any interior point such that OA
= OC. What is five-ninths of the degree measure of ∠DOB?Solution
ABCD is a rhombus. O is an interior point such that OA = OC. We know: In a rhombus, diagonals bisect each other at 90°. So, ∠AOC = ∠BOD = 90°. But here, OA = OC, which means point O lies on the perpendicular bisector of AC, i.e., O lies on diagonal BD. So now, in triangle DOB: Since O lies on BD (diagonal), and OA = OC, that forces O to be on BD, making triangle DOB an isosceles right triangle. Then, ∠DOB = 180°, because now O lies on diagonal BD, and DOB becomes a straight angle. Now (5/9) × 180° = 100°
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