Question

8. ABCD is a quadrilateral in which AD = BC.
E, F, G and H are the mid-points of AB, BD,
CD and AC respectively. Prove that EFGH is
a rhombus
​

Answer 1

Qᴜᴇsᴛɪᴏɴ -

ABCD is a quadrilateral in which AD = BC.

E, F, G and H are the mid-points of AB, BD, CD and AC respectively. Prove that EFGH is a rhombus.

ɢɪᴠᴇɴ -

• ABCD is a quadrilateral.
• AD = BC.
• E,F,G and H are the midpoints of AB, BD, CD and AC respectively.

Tᴏ ᴘʀᴏᴠᴇ -

• EFGH is a parallelogram.

ᴘʀᴏᴏғ -

AD = BC ________(given)

now,

GH || AD ________(mid-point theorem)

∴ GH = 1/2 AD _______(i)

And,

EF || AD ________(mid-point theorem)

∴ EF = 1/2 AD _______(ii)

From eq. (i) and (ii),

⇰GH || EF

Now,

GH || EF ________(mid-point theorem)

∴ GF = 1/2 BC _______(iii)

And,

EH || BC ________(mid-point theorem)

∴ EH = 1/2 BC ________(iv)

From eq. (iii) and (iv),

⇰GF || EH

∴ EFGH is a parallelogram. (opposite sides are parallel)

since,

EF = GH (adjacent sides of parallelogram are equal)

∴ EF = FG = GH = HE

∴ EFGH is a rhombus.

______________________(proved)

@MissSolitary✌️

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