Question

AD and BC are equal perpendiculars to a line segment AB . show that CD bisects AB. ( by using a criteria)

Answer 1

In triangle OAD & OBC
AD=BC (GIVEN)
<OAD=<OBC (EACH=90*)
<AOD=<BOC (VERTICLE OPP ANGLE)
THEREFORE,
TRIANGLE ODA=OCB (AAS CRITERIA)

OA=OB (CPCT)

So, CD bisects AB.

Answer 2

$\bf{\underline{Q.\: AD\: and \:BC \:are\: perpendiculars\:}}$ $\bf{\underline{to\: a\: line\: segment\: AB. \:Show\: that\: }}$
$\bf{\underline{ CD\:bisects\: AB.}}$

$\mathbb{ SOLUTION }$:

Given,
AD and BC are perpendiculars of AB.

$\mathbb{ TO \:BE \:SHOWN }$:
CD bisects AB

∠BOC = ∠DOC ( ∴ Vertically opposite angles )

DA = BC
∠B = ∠A = 90°

So, by $\bf{AAS}$ congruence condition, ΔBOC ≅ ΔOAD

So,
now CO = OD
so, it bisects AB on point '$\bf{O}$'

OA = OB [ by $\bf{CPCT}$ ]

➖➖➖➖➖➖➖➖➖➖➖➖➖

$\bf{AAS}$ : Angle-Angle-side

$\bf{CPCT}$ : Corresponding Parts of Congruent Triangles