Question

In the given figure, if AOB is a line segment and ∠GOC = ∠HOD = ∠IOE = ∠FOB = 90°, then which of the following is always true?

• A. $\frac{x+c+a}{b+y+z}=1$
  
• B. $x+c+b+y=2a+2z$
• C. $x+c+b+y+a+z=360°$
• D. $\frac{x+y+b+c}{a+z}=3$
  

Answer 1

The correct option is B $x+c+b+y=2a+2z$

Given: ∠GOC = ∠HOD = ∠IOE = ∠FOB = 90°

Since AOB is a straight line.
∴ ∠AOI + ∠IOE + ∠EOB = 180° (Linear pair)
⇒ a + ∠IOE + z = 180°
⇒ a + z = 180° – 90° (∵ ∠IOE = 90°)
⇒ a + z = 90° …..(i)
Similarly, b + y = 90° …..(ii)
and, c + x = 90° …..(iii)
Adding (ii) and (iii), we get
b + y + c + x = 90° + 90°
⇒ x + c + b + y = 180°
⇒ x + c + b + y = 2 × 90°
⇒ x + c + b + y = 2(a + z) [From (i)]
⇒ x + c + b + y = 2a + 2z
Hence, the correct answer is option (b).