Question

Find the value of 'x' and 'y'

Answer 1

GIVEN: A triangle ABC, AD = BC,

BD = DC , SO, Angle DBC = Angle DCB = 2x ( angles opposite to equal sides of a triangle)

Angle BAD = 3x & ABD = y

TO FIND: the value of x

PROOF & CALCULATION:

In triangles DAB ,

AD/ sin y = BD/ sjn3x

=> AD/BD = sin y / sin 3x

=> AD/ BD = sin( 180 - 7x) / sin3x

= AD / BD = sin 7x / sin 3x …………(1)

Now, in triangle DBC

BC / sin( 180–4x) = BD/ sin 2x

=> BC/ sin 4x = BD / sin 2x

=> AD/ sin 4x = BD/ sin 2x

=> AD / BD = sin 4x / sin 2x ……………..(2)

By (1) & (2)

Sin 7x / sin 3x = sin 4x / sin 2x

=> sin 7x * sin 2x = sin 3x * sin 4x

=> sin 7x * sin 2x = sin 3x * sin 2*2x

=> sin 7x* sin2x = sin 3x * 2sin 2x cos 2x

= sin 7x = 2 sin 3x cos 2x

Now, since, 2 sin A cos B = sin( A+ B)+ sin( A-B)

=> sin 7x = 2 sin 3x cos 2x = sin( 3x + 2x) + sin ( 3x - 2x)

=> sin 7x = sin 5x + sin x

=> sin x = sin 7x - sin 5x

=> sin x = sin (6x + x) - sin ( 6x - x)

=> sin x = 2 sin x * cos 6x

=> 1 = 2 * cos 6x

=> cos 6x = 1/2

=> 6x = 60°

=> x = 10°

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Answer 2

so y= 60 degree and x= 30 degree explained in the photo