Question

Draw a circle of radius 4 cm and construct a chord of 6 cm length in it. Draw two angles in major segment and two angles in minor segment. Verify that angles in major segment are acute angles and angles in minor segment are obtuse angles by measuring them.

Answer 1

Draw a circle of radius $r=4cm$. Mark a chord of length $6cm$

Take two points $A$ and $B$ in major sector and two points $C$ and $D$ in minor sector.

Join $PQ$ with $A,B,C$ and $D$

$\angle POQ=2\angle PAQ\angle POQ=2\angle PBQ$ (half angle property)

$\angle PAQ=\angle PBQ$

Also, $0<\angle POQ<180$

Gives, $0<\angle PBQ<900<\angle PAQ<90$ (acute angle)

$PBQC,PBQD,PAQC,PAQD$ are cyclic quadrilateral

i.e. $\angle PBQ+\angle PDQ={180}^o\angle PBQ+\angle PCQ={180}^o\angle PAQ+\angle PDQ={180}^o\angle PAQ+\angle PCQ={180}^o$

$0<\angle PBQ<90\Rightarrow 90<\angle PDQ<18090<\angle PCQ<{180}^o$ (obtuse angle)

$0<\angle PAQ<90\Rightarrow 90<\angle PDQ<18090<\angle PCQ<{180}^o$ (obtuse angle)