In the adjoining figure, CD is a diameter of the circle with centre O.

1.	In the adjoining figure, CD is a diameter of the circle with centre O. Diameter CD is perpendicular to chord AB at point E. Show that ∆ABC is an isosceles triangle.Given: O is the centre of the circle.Diameter CD ⊥ chord AB, A-E-B To prove: ∆ABC is an isosceles triangle.
Answer» Proof: diameter CD ⊥ chord AB [Given] ∴ seg OE ⊥ chord AB [C-O-E, O-E-D] ∴ seg AE ≅ seg BE ……(i) [Perpendicular drawn from the centre of the circle to the chord bisects the chord] 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.

In the adjoining figure, CD is a diameter of the circle with centre O. Diameter CD is perpendicular to chord AB at point E. Show that ∆ABC is an isosceles triangle.Given: O is the centre of the circle.Diameter CD ⊥ chord AB, A-E-B To prove: ∆ABC is an isosceles triangle.

Answer»

Proof:

diameter CD ⊥ chord AB [Given]

∴ seg OE ⊥ chord AB [C-O-E, O-E-D]

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

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.

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