The mid-points of the sides of a triangle along with any of the vertices as the fourth point make a parallelogram of area equal to 1. area of △ABC 2. area of △ABC 3. area of △ABC 4. area of △ABC

Question

KnowledgeBoat · Accepted Answer

Let CDEF be a parallelogram. So, diagonal DF divides it into two triangles of equal area. ∴ area of △CDF = area of △EDF ........(i) E and F are mid-points of side AB and AC respectively. By mid-point theorem, EF = BC = BD (As D is mid-point of BC) and EF || BC. Since, FE || BC so from figure, EF || BD and EF = BD. So, EBDF is a paralleogram with diagonal ED dividing it into two triangles of equal area. ∴ area of △EBD = area of △EDF ........(ii) E and D are mid-points of side AB and BC respectively. By mid-point theorem, ED = AC = AF (As F is mid-point of AC) and ED || AC. Since, ED || AC so from figure, ED || AF. So, AEDF is a paralleogram with diagonal EF dividing it into two triangles of equal area. ∴ area of △AEF = area of △EDF ........(iii) From (i), (ii) and (iii) we get, area of △EDF = area of △CDF = area of △EBD = area of △AEF = x From figure, area of △ABC = area of △EDF + area of △CDF + area of △EBD + area of △AEF = 4x. area of || gm EDCF = area of △EDF + area of △CDF = 2x. So, area of || gm EDCF = area of △ABC. Hence, Option 1 is the correct option.

Mathematics

The mid-points of the sides of a triangle along with any of the vertices as the fourth point make a parallelogram of area equal to 1. area of △ABC 2. area of △ABC 3. area of △ABC 4. area of △ABC

Theorems on Area

Related Questions

In the adjoining figure, area of parallelogram ABCD is
AB × BM
BC × BN
DC × DL
AD × DL

In the adjoining figure, AB || DC and AB ≠ DC. If the diagonals AC and BD of the trapezium ABCD intersect at O, then which of the following statements is not true ?
area of △ABC = area of △ABD
area of △ACD = area of △BCD
area of △OAB = area of △OCD
area of △OAD = area of △OBC

In the adjoining figure, ABCD is a trapezium with parallel sides AB = a cm and DC = b cm. E and F are mid-points of the non-parallel sides. The ratio of area of ABFE and area of EFCD is
a : b
(3a + b) : (a + 3b)
(a + 3b) : (3a + b)
(2a + b) : (3a + b)

A median of a triangle divides it into two
triangles of equal area
congruent triangles
right triangles
isosceles triangles

Mathematics

The mid-points of the sides of a triangle along with any of the vertices as the fourth point make a parallelogram of area equal to 1. area of △ABC 2. area of △ABC 3. area of △ABC 4. area of △ABC

Theorems on Area

Related Questions

In the adjoining figure, area of parallelogram ABCD isAB × BMBC × BNDC × DLAD × DL

In the adjoining figure, AB || DC and AB ≠ DC. If the diagonals AC and BD of the trapezium ABCD intersect at O, then which of the following statements is not true ?area of △ABC = area of △ABDarea of △ACD = area of △BCDarea of △OAB = area of △OCDarea of △OAD = area of △OBC

In the adjoining figure, ABCD is a trapezium with parallel sides AB = a cm and DC = b cm. E and F are mid-points of the non-parallel sides. The ratio of area of ABFE and area of EFCD isa : b(3a + b) : (a + 3b)(a + 3b) : (3a + b)(2a + b) : (3a + b)

A median of a triangle divides it into twotriangles of equal areacongruent trianglesright trianglesisosceles triangles

In the adjoining figure, area of parallelogram ABCD is
AB × BM
BC × BN
DC × DL
AD × DL

In the adjoining figure, AB || DC and AB ≠ DC. If the diagonals AC and BD of the trapezium ABCD intersect at O, then which of the following statements is not true ?
area of △ABC = area of △ABD
area of △ACD = area of △BCD
area of △OAB = area of △OCD
area of △OAD = area of △OBC

In the adjoining figure, ABCD is a trapezium with parallel sides AB = a cm and DC = b cm. E and F are mid-points of the non-parallel sides. The ratio of area of ABFE and area of EFCD is
a : b
(3a + b) : (a + 3b)
(a + 3b) : (3a + b)
(2a + b) : (3a + b)

A median of a triangle divides it into two
triangles of equal area
congruent triangles
right triangles
isosceles triangles