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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. |
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Answer» 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 ……(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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