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Answers
x+2y=90 is the Desired Operation
Step-by-step explanation:
Given-
Given-QRP = y
Given-QRP = yORP = 90 (radius to tangent through pt of contact)
ORP - QRP = 90 - y = QRO
ORP - QRP = 90 - y = QRO
ORP - QRP = 90 - y = QRO now,
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = y
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVED
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)ORS = OSR = y (ANGLES OPP.TO EQUAL SIDES-RADII)
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)ORS = OSR = y (ANGLES OPP.TO EQUAL SIDES-RADII)QOR=2 OSR (ANGLE SUBTENTED AT THE CENTRE BY A CHORD IS TWICE THE ANGLE SUBTENTED BY IT ON ANY OTHER PT ON THE CIRCLE)
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)ORS = OSR = y (ANGLES OPP.TO EQUAL SIDES-RADII)QOR=2 OSR (ANGLE SUBTENTED AT THE CENTRE BY A CHORD IS TWICE THE ANGLE SUBTENTED BY IT ON ANY OTHER PT ON THE CIRCLE)QOR = 2y
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)ORS = OSR = y (ANGLES OPP.TO EQUAL SIDES-RADII)QOR=2 OSR (ANGLE SUBTENTED AT THE CENTRE BY A CHORD IS TWICE THE ANGLE SUBTENTED BY IT ON ANY OTHER PT ON THE CIRCLE)QOR = 2yOQR = QPR+QRP (EXT. ANGLE PROP)
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)ORS = OSR = y (ANGLES OPP.TO EQUAL SIDES-RADII)QOR=2 OSR (ANGLE SUBTENTED AT THE CENTRE BY A CHORD IS TWICE THE ANGLE SUBTENTED BY IT ON ANY OTHER PT ON THE CIRCLE)QOR = 2yOQR = QPR+QRP (EXT. ANGLE PROP)OQR = x+y
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)ORS = OSR = y (ANGLES OPP.TO EQUAL SIDES-RADII)QOR=2 OSR (ANGLE SUBTENTED AT THE CENTRE BY A CHORD IS TWICE THE ANGLE SUBTENTED BY IT ON ANY OTHER PT ON THE CIRCLE)QOR = 2yOQR = QPR+QRP (EXT. ANGLE PROP)OQR = x+yOQR = ORQ = x+y (ANGLES OPP. TO EQUAL SIDES)
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)ORS = OSR = y (ANGLES OPP.TO EQUAL SIDES-RADII)QOR=2 OSR (ANGLE SUBTENTED AT THE CENTRE BY A CHORD IS TWICE THE ANGLE SUBTENTED BY IT ON ANY OTHER PT ON THE CIRCLE)QOR = 2yOQR = QPR+QRP (EXT. ANGLE PROP)OQR = x+yOQR = ORQ = x+y (ANGLES OPP. TO EQUAL SIDES)OQR + ORQ + QOR = 180 (ANGLE SUM PROP)
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)ORS = OSR = y (ANGLES OPP.TO EQUAL SIDES-RADII)QOR=2 OSR (ANGLE SUBTENTED AT THE CENTRE BY A CHORD IS TWICE THE ANGLE SUBTENTED BY IT ON ANY OTHER PT ON THE CIRCLE)QOR = 2yOQR = QPR+QRP (EXT. ANGLE PROP)OQR = x+yOQR = ORQ = x+y (ANGLES OPP. TO EQUAL SIDES)OQR + ORQ + QOR = 180 (ANGLE SUM PROP)x + y + x + y + 2y = 180
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)ORS = OSR = y (ANGLES OPP.TO EQUAL SIDES-RADII)QOR=2 OSR (ANGLE SUBTENTED AT THE CENTRE BY A CHORD IS TWICE THE ANGLE SUBTENTED BY IT ON ANY OTHER PT ON THE CIRCLE)QOR = 2yOQR = QPR+QRP (EXT. ANGLE PROP)OQR = x+yOQR = ORQ = x+y (ANGLES OPP. TO EQUAL SIDES)OQR + ORQ + QOR = 180 (ANGLE SUM PROP)x + y + x + y + 2y = 1802x + 4y = 180
ORP - QRP = 90 - y = QRO now,QRS = 90° (ANGLE IN SEMICIRCLE)QRO + ORS = QRS(90-y) + ORS = 90ORS = 90-(90-y)ORS = yHENCE PROVEDii)ORS = OSR = y (ANGLES OPP.TO EQUAL SIDES-RADII)QOR=2 OSR (ANGLE SUBTENTED AT THE CENTRE BY A CHORD IS TWICE THE ANGLE SUBTENTED BY IT ON ANY OTHER PT ON THE CIRCLE)QOR = 2yOQR = QPR+QRP (EXT. ANGLE PROP)OQR = x+yOQR = ORQ = x+y (ANGLES OPP. TO EQUAL SIDES)OQR + ORQ + QOR = 180 (ANGLE SUM PROP)x + y + x + y + 2y = 1802x + 4y = 180x + 2y = 90 is the desired operation !
OR
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