In ABC, I is the incentre. If BIC : CIA : AIB= 5 : 6 : 7, then the value of A : B : C is(1) 3 : 1 : 5 (2) 1 : 3 : 5(3) 5 : 3 : 1 (4) 1 : 5 : 3
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are given values areas or angles ?
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Look at the diagram.
ID, IE, IF are radii of incircle and perpendiculars on to the sides from incenter I.
Triangles AIE, AIF are similar as AI is common, IE=IF, and angles IEA = IFA = 90 deg. Also, AE=AF as tangents from same point A to incircle.
Similarly, triangles BIF, BID are similar. Also, CID, and CIE are similar.
Hence, angle AIF=AIE= z.
In the quadrilateral AFIE, angle A = 360-(90+90+FIE) = 180 - 2 z
Similarly from quadrilateral BFID, we get angle B = 180 - 2 x
From quadrilateral CDIE, we get angle C = 180 - 2 y
We are given that BIC : CIA : AIB = 5 : 6 : 7
x+y : y+z : z+x = 5 : 6 : 7
(x+y)6 = (y+z)5 => y + 6x = 5z
(x+y)7 = (z+x) 5 => 2x + 7y = 5z
solving these simultaneous equations, we get 6 y = 4 x, or 3 y = 2 x
and 10 y = 5z, z = 2y
angle A = 180 - 2 z = 180 - 4y, angle B = 180 - 2 x = 180 - 3y
angle C = 180 - 2 y
Their sum = A+B+C = 180 => 9 y = 360, y = 40 deg.
angle A = 20 deg, angle B = 60 deg angle C = 100 deg.
A : B : C = 1 : 3 : 5
ID, IE, IF are radii of incircle and perpendiculars on to the sides from incenter I.
Triangles AIE, AIF are similar as AI is common, IE=IF, and angles IEA = IFA = 90 deg. Also, AE=AF as tangents from same point A to incircle.
Similarly, triangles BIF, BID are similar. Also, CID, and CIE are similar.
Hence, angle AIF=AIE= z.
In the quadrilateral AFIE, angle A = 360-(90+90+FIE) = 180 - 2 z
Similarly from quadrilateral BFID, we get angle B = 180 - 2 x
From quadrilateral CDIE, we get angle C = 180 - 2 y
We are given that BIC : CIA : AIB = 5 : 6 : 7
x+y : y+z : z+x = 5 : 6 : 7
(x+y)6 = (y+z)5 => y + 6x = 5z
(x+y)7 = (z+x) 5 => 2x + 7y = 5z
solving these simultaneous equations, we get 6 y = 4 x, or 3 y = 2 x
and 10 y = 5z, z = 2y
angle A = 180 - 2 z = 180 - 4y, angle B = 180 - 2 x = 180 - 3y
angle C = 180 - 2 y
Their sum = A+B+C = 180 => 9 y = 360, y = 40 deg.
angle A = 20 deg, angle B = 60 deg angle C = 100 deg.
A : B : C = 1 : 3 : 5
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