If E, F, G and H are mid-points of the sides AB, BC, CD and DA, respectively of a parallelogram ABCD, prove that area of the quad. EFGH = area of || gm ABCD.

Question

KnowledgeBoat · Accepted Answer

Parallelogram ABCD with E, F, G and H as mid-points of the sides AB, BC, CD and DA, respectively is shown below: AH = AD (As H is the mid-point of AD) BF = BC (As F is the mid-point of BC) So, AH = BF and AH || BF (As AD || BC). So, ABFH is a || gm. Since, || gm ABFH and △EFH are on same base FH and between same parallel lines AB and FH so, area of △EFH = area of || gm ABFH .......(i) Similarly, HD = AD (As H is the mid-point of AD) FC = BC (As F is the mid-point of BC) So, HD = FC and HD || FC (As AD || BC). So, HFCD is a || gm. Since, || gm HFCD and △HFG are on same base HF and between same parallel lines HF and DC so, area of △HFG = area of || gm HFCD .......(ii) Adding (i) and (ii) we get, area of △EFH + area of △HFG = area of || gm ABFH + area of || gm HFCD area of quad. EFGH = (area of || gm ABFH + area of || gm HFCD) = (area of || gm ABCD). Hence, proved that area of the quad. EFGH = area of || gm ABCD.

Mathematics

If E, F, G and H are mid-points of the sides AB, BC, CD and DA, respectively of a parallelogram ABCD, prove that area of the quad. EFGH = area of || gm ABCD.

Theorems on Area

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Mathematics

If E, F, G and H are mid-points of the sides AB, BC, CD and DA, respectively of a parallelogram ABCD, prove that area of the quad. EFGH = area of || gm ABCD.

Theorems on Area

Related Questions

In figure (1) given below, ABCD is a parallelogram and P is any point in BC. Prove that, Area of ∆ABP + area of ∆DPC = Area of ∆APD.

In figure (1) given below, ABCD is a parallelogram. P, Q are any two points on the sides AB and BC respectively. Prove thatarea of ∆CPD = area of ∆AQD.

In figure (1) given below, ABCD is a parallelogram. P, Q are any two points on the sides AB and BC respectively. Prove that
area of ∆CPD = area of ∆AQD.