Question

construct an angle of 90° at the initial point of a given ray and justify the construction​

Answer 1

Solution

Steps of construction 

Draw a line segment OA.Taking O as center and any radius, draw an arc cutting OA at B.Now, taking B as center and with the same radius as before, draw an arc intersecting the previously drawn arc at point C.With C as center and the same radius, draw an arc cutting the arc at D.With C and D as center and radius more than 21CD, draw two arc intersecting at P.Join OP.Thus, ∠AOP=90o

Justification 

Join OC and BC

Thus,

OB=BC=OC             [Radius of equal arcs]

∴△OCB is an equilateral triangle

∴∠BOC=60o

Join OD,OC and CD

Thus,

OD=OC=DC             [Radius of equal arcs]

∴△DOC is an equilateral triangle

∴∠DOC=60o

Join PD and PC

Now,

In △ODP and △OCP

OD=OC             [Radius of same arcs]

DP=CP         [Arc of same radii]

OP=OP        [Common]

∴△ODP≅△OCP       [SSS congruency]

∴∠DOP=∠COP    [CPCT]

So, we can say that

∠DOP=∠COP=21∠DOC

∠DOP=∠COP=21×60=30o

Now,

∠AOP=∠BOC+∠COP

∠AOP=60+30

∠AOP=90o

Hope it helps ❤