Question

AB and AC are two chords of a circle and the diameter passing through A bisect the angle angle BAC .prove that AB=AC

Answer 1

Answer:

hope it helps.

Step-by-step explanation:

Given: AB and CD are two equal chord of the circle.

Prove: Center O lies on the bisector of the ∠BAC.

Construction: Join BC. Let the bisector on ∠BAC intersect BC in P.

Proof:

In△APB and △APC

AB=AC      ...(Given)

∠BAP=∠CAP        ...(Given)

AP=AP            ....(common)

∴△APB≅△APC      ...SAS test

⇒BP=CP  and  ∠APB=∠APC        ...CPCT

∠APB+∠APC=180  ...(Linear pair)

∴2∠APB=180  ....(∠APB=∠APC)

⇒∠APB=90  

∴AP is perpendicular bisector of chord BC.

Hence, AP passes through the center of the circle.