# In triangle abc = 90 degree o is the midpoint of ac show that oa = ob = oc = ac by 2

2
by eswaryogi

2014-03-27T08:08:20+05:30
Constrution. Join OA.
Given:-
O is midpoint of AC
this imply OA=OC (Midpint divides lig segment in equal halfs)
now,In Trianle ABC abd OBC
Angle "A" = Angle "A"( common)
Anglle "C" = Angle "C"(common)
OA = OC ( equal).
Now,
BC   =     AB   =  BC
BC
OB       BC
now,
AB * Bc = Bc* Ob
or
AB = OB

......

2014-03-29T21:05:40+05:30
At first draw the figure.
now draw a line parallel to BC.
now draw another line parallel to AB.
let us name the point at which these two lines intersect each other as D.
join A and D.
now the quadrilateral ABCD becomes a rectangle since its opposite sides(AD,BC and AB,DC) are equal and are parallel to each other.
now let us consider the two triangles,BCD and BAD.
here,
CD = BA.
by SAS congruency rule, we can conclude that the triangles BCD and BAD are congruent to each other.
we know that for any two congruent triangles all the corresponding parts are equal to each other.
so now,
angle DBC = angle ABD.
here the magnitudes of the four angles is half of 90 degrees as they are equal to each other. since half of 90 degrees is 45 degrees, the magnitude of the four angles are equal to 45 degrees.
now let us consider the triangle BAD.
here,
angle ADB = angle DBA = 45 degrees.
we know that in a triangle if two angles are equal, then their corresponding opposite angles are equal to each other.
here,
AB and AD  are corresponding opposite sides of the two equal angles. so AB = AD.
as a result all the sides of the rectangle become equal.
now consider the two triangles ADC and ABC.
here,
BC = CD.