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  1. Chapter 11 Class 9 Constructions
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Construct an angle of 45 at the initial point of a given ray and justify the construction . Steps of construction Draw a ray OA. Taking O as centre and any radius, draw an arc cutting OA at B. Now, taking B as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at point C. With C as centre and the same radius, draw an arc cutting the arc at D. With C and D as centres and radius more than 1/2 CD draw two arcs intersecting at P. Join OP. Thus, AOP = 90 Now, take B and Q as centers, and radius greater than 1/2 BQ, draw two arcs intersecting at R. Join OR. Thus, AOR = 45 Justification We need to prove AOR = 45 Join OC Thus, OB = BC = OC OCB is an equilateral triangle BOC = 60 Join OD, OC and CD Thus, OD = OC = DC DOC is an equilateral triangle DOC = 60 Join PD and PC Now, In ODP and OCP OD = OC DP = CP OP = OP ODP OCP DOP = COP So, we can say that DOP = COP = 1/2 DOC DOP = COP = 1/2 60 = 30 Now, AOP = BOC + COP AOP = 60 + 30 = 90 Now, Join QR and BR In OQR and OBR OQ = OB QR = BR OR = OR OQR OBR QOR = BOR QOR = BOR = 1/2 AOP DOP = COP = 1/2 90 = 45 Thus, AOR = 45 Hence justified

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.