Show that the image formed in a plane mirror is as far behind the mirror as the object is in front

Show that the image formed in a plane mirror is as far behind the mirror as the object is in front

Consider an object A placed in front of a mirror M.

A ray AD from A incident normally on the mirror at D is reflected back along DA.

The  Ray AO is reflected along OB and appears to come from point I behind the mirror.

The intersection I of the rays AD and BO is the image position.

From above angles,

x = i                  (alternating angles)

i = r                  (2nd law of reflection)

r = y                 (corresponding angles)

Combining all the equations gives

x = i = r = y

Þ   angle x = angle y

Tan x   =  tan y

DO/AD = DO/ID

Thus  AD  =  ID.

Therefore, the image is as far behind the mirror as the object is in front

Thank you

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