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  • AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters,

Q 7. AC and BD are chords of a circle that bisect each other. Prove that

(i) AC and BD are diameters

Answers (1)

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Given: AC and BD are chords of a circle which bisect each other.

To prove: AC and BD are diameters.

Construction: Join AB,BC,CD,DA. 

Proof :

In △ABD and △CDO,
AO=OC-----(Given)\angle \mathrm{AOB}=\angle \mathrm{COD}−−−−(Verticallyoppositeangles)\mathrm{BO}=\mathrm{DO}−−−−(Given)So,\triangle \mathrm{ABD} \cong \triangle \mathrm{CDO}(BySAS)\angle \mathrm{BAO}=\angle \mathrm{DCO} 
 ---(\mathrm{CPCT})\angle \mathrm{BAO}and\angle \mathrm{DCO}arealternateanglesandareequal.So,A B \| D C−−−−−−−−−−1AlsoA D \| B C$ ---------- 2

From 1 and 2,

∠A+∠C=180∘
∠A=∠C
From 3 and 4,
∠A+∠A=180∘
⇒2∠A=180∘
⇒∠A=90∘

BD is the diameter of the circle.

Similarly, AC is a diameter.

Posted by

seema garhwal

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