Question

The diagonal of a rhombus ABCD intersect at 0.If AC = 40cm and BD = 42cm,find the length of the side of rhombus
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Answer 1

$\LARGE\mathfrak{\underline{\underline{ Given :-}}}$

$\qquad\tt{:}\longrightarrow\large\texttt{Length of AC = 40cm}$

$\qquad\tt{:}\longrightarrow\large\texttt{Length of BD = 42cm }$

$\LARGE\mathfrak{\underline{\underline{To \: \: \: Find :-}}}$

$\qquad\tt{:}\longrightarrow\large\texttt{Length of the side of the rhombus = ? }$

$\LARGE\mathfrak{\underline{\underline{How \; To \; Solve :-}}}$

• As per the properties of the rhombus, it's diagonals bisect each other at a angle of 90° .

• As the diagonals bisect each other, so the diagonals are divided into two equal parts .

$\LARGE\mathfrak{\underline{\underline{Solution :-}}}$

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↦ As the diagonals are divided into two equal parts

• So ,

• AO = 1/2 × 40 = 20 cm
• BO = 1/2 × 42 = 21 cm

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➜ In ∆ AOB

• ∠ AOB = 90° ..... ( As per the properties of the rhombus )

↦ So ∆ AOB is a Right Angled Triangle .

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$\qquad\tt{:}\longrightarrow\large\textsf{By using Pythagoras Theorem :- }$

$\qquad\tt{:}\longrightarrow\large\textsf{AO² + BO² = AB²}\\\\\\\qquad\tt{:}\longrightarrow\large\textsf{20² + 21² = AB² }\\\\\\\qquad\tt{:}\longrightarrow\large\textsf{400 + 441 = AB² }\\\\\\\qquad\tt{:}\longrightarrow\boxed{\large\textsf\textcolor{purple}{ 841 = AB² }}$

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↦ Taking Square Root on both the side's :-

$\qquad\tt{:}\longrightarrow\large\textsf{√841 = √AB²}\\\\\\\qquad\tt{:}\longrightarrow\boxed{\underline{\overline{\large\textsf\textcolor{red}{29 = AB}}}}$

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∴ The side of the Rhombus is 29 cm

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↦ What is a Rhombus :-

➛ Rhombus is a special case of a parallelogram, and it is a four-sided quadrilateral. In a rhombus, opposite sides are parallel and the opposite angles are equal. Moreover, all the sides of a rhombus are equal in length, and the diagonals bisect each other at right angles. The rhombus is also called a diamond or rhombus diamond.

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↦Properties of the Rhombus :-

• All sides of the rhombus are equal.

• The opposite sides of a rhombus are parallel.

• Opposite angles of a rhombus are equal.

• In a rhombus, diagonals bisect each other at right angles.

• Diagonals bisect the angles of a rhombus.

• The sum of two adjacent angles is equal to 180 degrees.

• The two diagonals of a rhombus form four right-angled triangles which are congruent to each other.

• You will get a rectangle when you join the midpoint of the sides.

• You will get another rhombus when you join the midpoints of half the diagonal.

• Around a rhombus, there can be no circumscribing circle.

• Within a rhombus, there can be no inscribing circle.

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