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Li ABC and Li DBC are two isosceles triangles on the same base BC and vertices A and D are on the same sid e of BC ( see 78 figure ). If AD is ertended to intersect BC at P, show that (i) fl ABD :: fl ACD (ii) flABP :: ti.ACP (iii) AP bisects L A as well as L D. (iv) AP is the perpendicular bisector of BC

Li ABC and Li DBC are two isosceles triangles on the same base BC and vertices A and D are on the same sid e of BC ( see


78
figure ). If AD is ertended to intersect BC at P, show that
(i) fl ABD :: fl ACD
(ii) flABP :: ti.ACP
(iii) AP bisects L A as well as L D.
(iv) AP is the perpendicular bisector of BC

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 8741 Points
6 months ago
Ci) Consider triangles ABD and ACD, We have AB = AC BD = CD AD = DA [Given] [Given] [Comm.on] So, ti.ABD :: ti.ACD [SSS rule] L BAD = L CAD .and LABD = LACD ...(i) [CPCT] (ii.) Consider triangles ABP and ACP, We have AB = AC AP = PA and L BAP = L CAP fl ABP ::: ti.ACP BP = PC, L BPA = L CPA [Given] [Comm.on] [From (i)] [SAS rule] (iii) L BAP = L CAP [From result (ii)] AP bisects LA Also, L BAD + L ABD = L CAD = L ACD [From (i)] L BDP = L CDP [Exterior angle property] So, DP bisects LD. Hence, AP bisects LA as well as LD. (iv) Also, L BPA + L CPA = 180° [Linear pair] 2L BPA = 180° As and L BPA = 90° BP = CP AP .i BC [From result (ii)] [From result (ii)] [Proved above] AP is perpendicular bisect.or of BC.

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