Question

In the given figure, O is the point of intersection of two chords AB and CD such that OB = OD and ∠AOC = 45°. Then, ΔОAC and ΔODB are ______.

Options

• equilateral and similar

• equilateral but not similar

• isosceles and similar

• isosceles but not similar

Answer 1

In the given figure, O is the point of intersection of two chords AB and CD such that OB = OD and ∠AOC = 45°. Then, ΔОAC and ΔODB are isosceles and similar.

Explanation:

In ΔΟAC and ΔODB, we have

∠AOC = ∠DOB (Vertically opposite angle) and OAC = ∠ODB (Angle in the same segment).

∴ ΔOAC ∼ ΔODB

⇒ `(OC)/(OB) = (OA)/(OD) = (AC)/(BD)`

∴ `(OC)/(OA) = (OB)/(OD) = 1`

⇒ OC = OA   ...[∵ OB = OD (given)]

Clearly, OA ≠ OD.

∴ `(OA)/(OD) ≠ 1`

⇒ `(AC)/(BD) ≠ 1`

⇒ AC ≠ BD

∴ ΔOAC and ΔODB are isosceles and similar.