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ABCD is a rectangle such that its diagonal BD bisects $∠ B$ . Then ABCD is a ____.

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Detailed Solution

ABCD is a rectangle such that its diagonal BD bisects

∠ B

. Then ABCD is a square.
We are given a rectangle ABCD in which diagonal BD bisects

∠ B

. We have to check what type of rectangle is ABCD.
Question Image

Show this by the congruence of triangles:
In,

Δ A B D

and

Δ C B D

∠ A B D = ∠ C B D

… (BD is the bisector of

∠ B

)

∠ A = ∠ C

... (angles of a rectangle are right-angled)

B D = B D

… (Common side)
Hence by the ASA postulate, both the triangles are congruent i.e.,

Δ A B D ≅ Δ C B D

.
Now, show that ABCD is a square:
As congruent triangles have equal sides and equal angles hence,

A B = B C, A D = C D

. … (i)
But in a rectangle, the opposite sides are equal,

A B = C D

and

A D = B C

… (ii)
Use equations (i) and (ii),

A B = B C = C D = D A

.
As all sides of the rectangle ABCD are equal, so it is a square.
Therefore, the required answer is square.

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