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.

By mid-point theorem, The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it. In △ ADC, G and H are mid-points of sides CD and AC respectively. ⇒ GH = ⇒ AD = 2GH .........(1) In △ ABD, E and F are mid-points of sides AB and BD respectively. ⇒ EF = ⇒ AD = 2EF .........(2) From equations (1) and (2), we get : ⇒ AD = 2GH = 2EF ........(3) In △ BCD, G and F are mid-points of sides CD and BD respectively. ⇒ GF = ⇒ BC = 2GF .........(4) In △ ABC, E and H are mid-points of sides AB and AC respectively. ⇒ EH = ⇒ BC = 2EH .........(5) From equations (4) and (5), we get : ⇒ BC = 2GF = 2EH ........(6) Given, ⇒ AD = BC .............(7) From equations (3), (6) and (7), we get : ⇒ 2GH = 2EF = 2GF = 2EH ⇒ GH = EF = GF = EH. Since, all sides of quadrilateral EFGH are equal. ∴ EFGH is a rhombus. Hence, proved that EFGH is a rhombus.

The diagonals of a quadrilateral intersect at right angles. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.

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L and M are the mid-points of sides AB and DC respectively of parallelogram ABCD. Prove that segments DL and BM trisect diagonal AC.

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In the given figure, l // m // n and D is mid-point of CE. If AE = 12.6 cm, then BD is :
12.6 cm
25.2 cm
6.3 cm
18.9 cm

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Mathematics

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.

Mid-point Theorem

Related Questions

The diagonals of a quadrilateral intersect at right angles. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.

L and M are the mid-points of sides AB and DC respectively of parallelogram ABCD. Prove that segments DL and BM trisect diagonal AC.

In the given figure, l // m // n and D is mid-point of CE. If AE = 12.6 cm, then BD is :
12.6 cm
25.2 cm
6.3 cm
18.9 cm

Mathematics

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.

Mid-point Theorem

Related Questions

The diagonals of a quadrilateral intersect at right angles. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.

L and M are the mid-points of sides AB and DC respectively of parallelogram ABCD. Prove that segments DL and BM trisect diagonal AC.

In the given figure, l // m // n and D is mid-point of CE. If AE = 12.6 cm, then BD is :12.6 cm25.2 cm6.3 cm18.9 cm

In the given figure, l // m // n and D is mid-point of CE. If AE = 12.6 cm, then BD is :
12.6 cm
25.2 cm
6.3 cm
18.9 cm