Find the area of the figure formed by joining the mid-points of the adjacent sides of a rhombus with diagonals 12 cm and 16 cm.
Theorems on Area
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Question

Find the area of the figure formed by joining the mid-points of the adjacent sides of a rhombus with diagonals 12 cm and 16 cm.

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Given: ABCD is a rhombus with diagonals 12 cm and 16 cm. Construction: Join the midpoints of AB, BC, CD and DA of the rhombus ABCD and name them M, N, O and P, respectively, to form the quadrilateral MNOP. Join the line PN. To Prove: MNOP is a parallelogram. Proof: The midpoint theorem states that the line segment joining the midpoints of two sides of a quadrilateral is parallel to the opposite side and half of its length. Since M and N are midpoints of AB and BC, by midpoint theorem: MN ∥ AC and MN = AC .................(1) Similarly, P and O are midpoints of AD and DC. By the midpoint theorem: PO ∥ AC and PO = AC .....................(2) From equations (1) and (2), we get: MN ∥ AC and MN = PO Now, M and P are midpoints of AB and AD. By the midpoint theorem: MP ∥ BD and MP = BD .................(3) Similarly, N and O are midpoints of BC and DC. By the midpoint theorem: NO ∥ BD and NO = BD .....................(4) From equations (3) and (4), we get: MP ∥ NO and MP = NO Thus, quadrilateral MNOP is a parallelogram. If a triangle and a parallelogram are on the same base and between the same parallels, the area of the triangle is equal to one-half area of the parallelogram. Ar(Δ MNP) = Ar(∥gm ABNP) .................(5) Ar(Δ PON) = Ar(∥gm PNCD) .................(6) Then area of rhombus ABCD = x d1 x d2 = x (12 x 16) cm2 = 96 cm2 Since the parallelogram MNOP consists of two such halves: Ar(MNOP) = (Ar(ABCD)) = x 96 = 48 cm2 Hence, the area of the figure formed = 48 cm2.

Mathematics

Find the area of the figure formed by joining the mid-points of the adjacent sides of a rhombus with diagonals 12 cm and 16 cm.

Theorems on Area

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Mathematics

Find the area of the figure formed by joining the mid-points of the adjacent sides of a rhombus with diagonals 12 cm and 16 cm.

Theorems on Area

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(i) the area of △APD
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