Question

The perimeter of a rhombus is 60 cm. If the length of longer diagonal is 24 cm,

find the length of shorter diagonal.​

Answer 1

Given :- ABCD is a rhombus with perimeter 60cm and length of its longer diagonal 24cm.

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❏ We need to find the length of shorter diagonal.

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Solution :- As we know that, all four sides of a rhombus are equal.

•°• AB = BC = CD = DA

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Now, perimeter of rhombus can be written as

☆ 4 × side

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→ Perimeter = 60

→ 4 × side = 60

→ side = 60/4

→ side = 15cm

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Therefore, all the sides of the rhombus is 15cm long.

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Since, diagonals of a rhombus are perpendicular bisector of each other.

•°• BD divides AC in two equal parts.

• CO = 24/2 = 12cm

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In right ∆BOC

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[By Pythagoras theorem]

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★ BC² = BO² + CO²

→ BO² = BC² - CO²

→ BO² = (15)² - (12)²

→ BO² = 225 - 144

→ BO² = 81

→ BO = √81

→ BO = 9 cm

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Since, AC also bisect BD, so we write that

→ BD = 2BO

→ BD = 2 × 9

→ BD = 18cm

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Hence,

• BD is the smaller diagonal with length of 18cm.

Answer 2

$\bold{Given}\begin{cases} \tt\: perimeter \: of \: a \: rhombus \: is = \bold{60cm} \\ \\ \tt \: length \: of \: the \: longer \: diagonal \: = \bold{24cm} \end{cases}$

We've to find the length of the shorter diagonal.

_________________________

As we know that :

$\underline{\boxed{\tt\purple{\bigstar \: perimeter \: of \: a \: rhombus \: = 4 \times side}}}$

Put the values -

$\tt:\implies \: perimeter \: of \: the \: rhombus \: = 4 \times side \\ \\ \\ \tt:\implies \: 60 = 4 \times side \\ \\ \\ \tt:\implies \: side = \cancel \frac{60}{4} \\ \\ \\ \tt:\implies \: side \: = \underline{\boxed{\sf\purple{15cm}}}\bigstar$

In ∆ BEC

$\tt:\implies \: BE² + EC² = BC² \: (diagonals \: of \: a \: rhombus \: bisect \: at \: right \: angles)$

Put the values -

$\tt:\implies \: BE {}^{2} + EC {}^{2} = BC {}^{2} \\ \\ \\ \tt:\implies \: (12) {}^{2} + EC {}^{2} = {(15)}^{2} \\ \\ \\ \tt:\implies \: 144 + EC {}^{2} \: = 225 \\ \\ \\ \tt:\implies \: EC {}^{2} \: = 22 5 - 144 \\ \\ \\ \tt:\implies \: EC {}^{2} \: = 81 \\ \\ \\ \tt:\implies \: EC \: = \sqrt{81} \\ \\ \\ \tt:\implies \: EC = \underline{\boxed{\tt\purple{9cm}}}\bigstar$

AC is the shorter diagonal diagonal. So,

$\tt:\implies \: AE + EC = AC \: \\ \\ \\ \tt:\implies \: EC + EC = AC \: (diagonals \: bisect \: each \: other \: ) \\ \\ \\ \tt:\implies \: AC = 9 + 9 \\ \\ \\ \tt:\implies \: AC = \underline{\boxed{\sf\purple{18cm}}}\bigstar \\ \\ \\ \sf\therefore{\underline{Hence, \: the \: shorter \: diagonal \: is \: \bold{18cm}.}}$