Question

Construct a cyclic quadrilateral $ABCD$ such that $AB=3.4cm,BC=4.2cm,AD=2.7cm$ and $\angle ABC={60}^{\circ }$. Then, $m\angle ADC$ (in degrees) $=?$

Answer 1

• Draw a line segment $BC$ of length $4.2cm$.
• Draw a line at an angle of ${60}^o$ by taking $BC$ as base and $B$ as vertex.
  
• Cut an arc of $3.4cm$ from $B$. This arc cuts the line at point $A$.
  
• Join $AB$.
  
• Draw perpendicular bisectors of $AB$ and $BC$. These perpendicular bisectors cuts each other at point $O$.
  
• Draw a circle of radius $OB$ and center $O$, this circle touches $A,B,C$.
  
• Mark an arc of length $2.7cm$ on circle from point $A$. This arc cuts the circle at point $D$.
  
• Join $ABCD$.
  
• $ABCD$ is the required cyclic quadrilateral.
  

$m\angle ADC={120}^o$ (measurement)

Also, $ABCD$ is a cyclic quadrilateral

$\angle ABC+\angle ADC={180}^o$ (opposite angle)

${60}^o+\angle ADC={180}^o\angle ADC={120}^o$