Question

Find the area of a rhombus whose perimeter is 80 m and one of whose diagonal is 24m

Answer 1

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Answer 2

$\star\;{\underline{\underline{\frak{AnswEr\;:}}}}$

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⠀

☯ Let ABCD be the rhombus of perimeter 80 m and diagonal AC = 24 m.

⠀⠀⠀ ⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

We have,

$:\implies\sf AB + BC + CD + DA = 80$

$:\implies\sf 4AB = 80\qquad\qquad\bigg\lgroup\bf \because\; AB = BC = CD = DA\bigg\rgroup$

$:\implies{\boxed{\sf{\pink{AB = 20 m}}}}\;\bigstar$

Now, In ∆ABC, we have

$:\implies\sf 2s = AB + BC + AC$

$:\implies\sf 2s = 20 + 20 + 24$

$\qquad:\implies\sf 2s = 64$

$\qquad:\implies{\boxed{\sf{\purple{s = 32}}}}\;\bigstar$

⠀⠀⠀ ⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━

⠀⠀⠀⠀⠀⠀⠀ ⠀☯ Now, Using Heron's Formula,

$\quad\quad\star\;{\boxed{\sf{\pink{\sqrt{s(s - a)(s - b)(s - c)}}}}}$

$:\implies\sf \sqrt{32(32 - 20)(32 - 20)(32 - 24)}$

$\qquad:\implies\sf \sqrt{32 \times 12 \times 12 \times 8}$

$\qquad\qquad:\implies\sf \sqrt{36864}$

$\qquad\qquad:\implies{\boxed{\sf{\purple{192\;m^2}}}}\;\bigstar$

$\therefore$ Hence, Area of rhombus ABCD = 2 × 192 m² = 384 m².