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(i) AB2 = BC . BD
(ii) AC2 = BC . DC
(iii) AD2 = BD . CD
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: (i) In ΔADB and ΔCAB, we have
∠DAB = ∠ACB (Each of 90o)
∠ABD = ∠CBA (Common angle)
Therefore,
ΔADB~ ΔCAB (AA similarity)
AB2 = CB * BD
(ii) Let ∠CAB = x
In ΔCBA,
∠CBA = 180o – 90o – x
∠CBA = 90o – x
Similarly, in ΔCAD
∠CAD = 90o - ∠CBA
= 90o – x
∠CDA = 180o – 90o – (90o – x)
∠CDA = x
In triangle CBA and CAD, we have
∠CBA = ∠CAD
∠CAB = ∠CDA
∠ACB = ∠DCA (Each 90o)
Therefore,
ΔCBA ~ΔCAD (By AAA similarity)
AC2 = DC * BC
(iii) In ΔDCA and ΔDAB, we have
∠DCA = ∠DAB (Each 90o)
∠CDA = ∠ADB (Common angle)
Therefore,
ΔDCA~ ΔDAB (By AA similarity)
AD2 = BD * CD
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