Question

In the given figure, CD is a diameter of the circle with centre O. Diameter CD is perpendicular to chord AB at point E. Show that ΔABC is an isosceles triangle.

Answer 1

Given: O is the centre of the circle.

Diameter CD ⊥ chord AB, A-E-B

To prove: ∆ABC is an isosceles triangle.

Proof:

Diameter CD ⊥ chord AB       ...(Given)

∴ seg OE ⊥ chord AB        ...(C-O-E, O-E-D)

∴ seg AE ≅ seg BE        ...(Perpendicular drawn from the centre of the circle to the chord bisects the chord)  ...(i)

In ∆CEA and ∆CEB,

∠CEA ≅ ∠CEB        ...(Each is of 90°)

seg AE ≅ seg BE       ...[From (i)]

seg CE ≅ seg CE      ...(Common side)

∴ ∆CEA ≅ ∆CEB      ...(SAS test)

∴ seg AC ≅ seg BC      ...(c.s.c.t.)

∴ ∆ABC is an isosceles triangle.