In the given figure, O is a point in the interior of square ABCD such

1.	In the given figure, O is a point in the interior of square ABCD such that △ OAB is an equilateral triangle. Show that △ OCD is an isosceles triangle.
Answer» We know that △ OAB is an equilateral triangle So it can be written as ∠ OAB = ∠ OBA = AOB = 60^o From the figure we know that ABCD is a square So we get ∠ A = ∠ B = ∠ C = ∠ D = 90^o In order to find the value of ∠ DAO We can write it as ∠ A = ∠ DAO + ∠ OAB By substituting the values we get 90^o = ∠ DAO + 60^o On further calculation ∠ DAO = 90^o – 60^o By subtraction ∠ DAO = 30^o We also know that ∠ CBO = 30^o Considering the △ OAD and △ OBC We know that the sides of a square are equal AD = BC We know that the sides of an equilateral triangle are equal OA = OB By SAS congruence criterion △ OAD ≅ △ OBC So we get OD = OC (c. p. c. t) Therefore, it is proved that △ OCD is an isosceles triangle.

In the given figure, O is a point in the interior of square ABCD such that △ OAB is an equilateral triangle. Show that △ OCD is an isosceles triangle.

Answer»

We know that △ OAB is an equilateral triangle

So it can be written as

∠ OAB = ∠ OBA = AOB = 60^o

From the figure we know that ABCD is a square

So we get

∠ A = ∠ B = ∠ C = ∠ D = 90^o

In order to find the value of ∠ DAO

We can write it as

∠ A = ∠ DAO + ∠ OAB

By substituting the values we get

90^o = ∠ DAO + 60^o

On further calculation

∠ DAO = 90^o – 60^o

By subtraction

∠ DAO = 30^o

We also know that ∠ CBO = 30^o

Considering the △ OAD and △ OBC

We know that the sides of a square are equal

AD = BC

We know that the sides of an equilateral triangle are equal

OA = OB

By SAS congruence criterion

△ OAD ≅ △ OBC

So we get OD = OC (c. p. c. t)

Therefore, it is proved that △ OCD is an isosceles triangle.