Question

In the figure, if x/3 = y/4 = z/5 calculate x, y and z (CBSE Grade 9, 2015)

Answer 1

Given,  

= x/3 = y/4 = z/5
i, e y =  z = 4x/3 and 5x/3

∵ A,B ,C and D forms a cyclic quadrilateral. 
 
∴ ∠ABC + ∠CDA = 180°....(1)

( Also, ∠CDQ + ∠CDA = 180°...(ii)

From (i) and (ii), ∠ABC = ∠CDQ

Also, ∠PCB = x... [Vertically opp angles Now,
 
In △ CDQ, x + y + ∠CDQ = 180° ⇒ x + z + ∠ABC = 180° ⇒ x + z + (180° − ∠PBC) = 180
 
⇒ x + z = ∠PBC

⇒ x + z = (180° − x − y)

⇒ 2x + y + z = 180°

⇒ 2x + + = 180° 4x/3 + 5x/3
 
⇒ 6x+4x+5x/3 = 180° 6x+4x+5x 3

⇒ 5x = 180°
 ⇒ x = 36° 

y = 4x/3 = 48° 

z = 5x/3 = 60° 

Answer 2

 x/3 = y/4 = z/5 

 A,B ,C and D forms a cyclic quadrilateral. 
 
 ∠ABC + ∠CDA = 180°_____1

( Also, ∠CDQ + ∠CDA = 180°________2

From (1) and (i2)
∠ABC = ∠CDQ 

Also, ∠PCB = x(Vertically opp angles)
 
In △ CDQ, x + y + ∠CDQ = 180°
⇒ x + z + ∠ABC = 180°
⇒ x + z + (180° − ∠PBC) = 180
 
⇒ x + z = ∠PBC 

⇒ x + z = (180° − x − y) 

⇒ 2x + y + z = 180° 

⇒ 2x + + = 180° 4x/3 + 5x/3
 
⇒ 6x+4x+5x/3 = 180° 6x+4x+5x 3 

⇒ 5x = 180°
 ⇒ x = 36° 

y = 4x/3 = 48° 

z = 5x/3 = 60°