In a rectangle ABCD, E, F, G, and H are the midpoints of the four sides, what kind of shape is represented by EFGH.
Answer:
Rhombus
- The following figure shows the rectangle ABCD,
Let's assume the length and breadth of the rectangle ABCD be a and b respectively. - In ΔGDH, DG =
=CD 2
,a 2
DH =
=DA 2
[Since, G and H are the midpoints of the sides CD and DA respectively.]b 2
∠D = 90° [Since, ABCD is a rectangle.]
GH2 = DG2 + DH2 [By the pythagorean theorem.]
⇒ GH2 = (
)2 + (a 2
)2b 2
Similarly, HE2 = EF2 = FG2 = (
)2 + (a 2
)2b 2
and hence, HE2 = EF2 = FG2 = GH2,
or HE = EF = FG = GH - We know that quadrilateral with all four sides equal is rhombus. Therefore, EFGH is a Rhombus.