Question

The area of a rhombus is equal to the area of a triangle. If the base of the triangle is 28 cm, its corresponding
altitude is 18 cm and one of the diagonals of a rhombus is 21 cm, find its other diagonal.​

Answer 1

${\underline{\underline{\bf{Given:}}}}$

• That, the area of a rhombus is equal to the area of a triangle.
• Base of the triangle = 28 cm. And It's corresponding height = 18 cm.
• Length of one diagonal of the rhombus = 21 cm.

${\underline{\underline{\bf{Need\:to\:find:}}}}$

• The length of another diagonal of the rhombus.

${\underline{\underline{\bf{Formulae\:to\:be\:used:}}}}$

$\underline{\boxed{\sf{{Area}_{(Triangle)}=\frac{1}{2}bh\:sq.units}}}$

$\underline{\boxed{\sf{{Area}_{(Rhombus)}=\frac{1}{2}{d}_{1}{d}_{2}\:sq.units}}}$

${\underline{\underline{\bf{Solution:}}}}$

According to the question,

$\rightarrow$ Area of triangle = Area of rhombus

$\rightarrow{\sf{\frac{1}{2}bh=\frac{1}{2}{d}_{1}{d}_{2}}}$

Now, by substituting the measures,

$\implies{\sf{\frac{1}{\cancel{2}}\times \cancel{28}\times 18=\frac{1}{2}\times 21\times {d}_{2}}}$

$\implies{\sf{14\times 18=\frac{1}{2}\times 21\times {d}_{2}}}$

$\implies{\sf{252=\frac{1}{2}\times 21\times {d}_{2}}}$

$\implies{\sf{252\times 2=1\times 21\times {d}_{2}}}$

$\implies{\sf{504=21\times {d}_{2}}}$

$\implies{\sf{\frac{504}{21}={d}_{2}}}$

$\implies{\sf{\red{\underline{\underline{24\:cm={d}_{2}}}}}}$

${\underline{\underline{\bf{Verification:}}}}$

$\rightarrow$ Area of triangle = Area of rhombus

$\rightarrow{\sf{\frac{1}{2}bh=\frac{1}{2}{d}_{1}{d}_{2}}}$

We know that the area of triangle is 252 m². So,

$\:\:\:\:\:\:\:\rightarrow{\sf{252=\frac{1}{2}\times 21\times \purple{24}}}$

$\:\:\:\:\:\:\:\rightarrow{\sf{252=\frac{1}{\cancel{2}}\times \cancel{504}}}$

$\:\:\:\:\:\:\:\rightarrow{\sf{\pink{252 = 252}}}$

$\:\:\:\:\:\:\:\rightarrow{\sf{L.H.S=R.H.S}}$

$\rightarrow$ So, our answer is correct.

${\underline{\underline{\bf{Required\:answer:}}}}$

• The length of second (another) diagonal of the rhombus is 24 cm.

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Answer 2

Correct Question-:

• The area of a rhombus is equal to the area of a triangle. If the base of the triangle is 28 cm, its corresponding altitude is 18 cm and one of the diagonals of a rhombus is 21 cm, find its other diagonal.

AnswEr- :

• $\underline{\boxed{\star{\sf{\blue{ Diagonal2 \: of \: Rhombus \:=24cm.}}}}}$

Explanation-:

$\frak{Given \:-:}\begin{cases} \sf{The \:area\: of \:a \:rhombus\: is\: equal \:to\: the\: area \:of\: a\: triangle .}& \\\\ \sf{The \:base \:of\: the \:triangle \:is \:28 \:cm,\:and \: its \:corresponding\:altitude\: is \:18 \:cm.} & \\\\\ \sf{ One\: of\: the\: diagonals \:of \:a \:rhombus \:is\: 21\: cm.} \end{cases}$

$\frak{To\: Find \:-:}\begin{cases} & \sf{The\:other \:Diagonal \:of\:Rhombus.} \end{cases}$

Now ,

• $\underline{\boxed{\star{\sf{\blue{ Area_{(Triangle)}\:= \frac{1}{2} × Base × Height.}}}}}$
• $\underline{\boxed{\star{\sf{\blue{ Area_{(Rhombus)}\:= \frac{1}{2} × Diagonal 1× Diagonal2.}}}}}$
• $\underline{\boxed{\star{\sf{\blue{ Area_{(Triangle)}\:=Area_{(Rhombus)}.}}}}}$

Here ,

• Base = 28 cm
• Height or Altitude = 18 cm
• Diagonal 1 = 21 cm
• Diagonal 2 = ?? .

Now ,

• $\implies{\sf{\large { \frac{1}{2} × 28 × 18 \: =\: \frac {1}{2} \times 21 \times D_{2}}}}$
• $\implies{\sf{\large { 28 × 9 \: =\: \frac {1}{2} \times 21 \times Diagonal_{2}}}}$
• $\implies{\sf{\large { 252 cm²\: =\: \frac {1}{2} \times 21 \times Diagonal_{2}}}}$
• $\implies{\sf{\large { 252 cm²\: =\: 10.5 \times Diagonal_{2}}}}$
• $\implies{\sf{\large { \frac {252}{10.5} \: =\: Diagonal_{2}}}}$
• $\implies{\sf{\large { 24 cm\: =\: Diagonal_{2}}}}$

Therefore,

• $\underline{\boxed{\star{\sf{\blue{ Diagonal2_{(Rhombus)}\:=24cm.}}}}}$
• $\underline{\boxed{\star{\sf{\blue{ Area_{(Rhombus)}\:=252cm².}}}}}$

Hence ,

• $\underline{\boxed{\star{\sf{\blue{ Diagonal2 \: of \: Rhombus \:=24cm.}}}}}$

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♤ Verification ♤

• $\underline{\boxed{\star{\sf{\blue{ Area_{(Rhombus)}\:= \frac{1}{2} × Diagonal 1× Diagonal2.}}}}}$

Here ,

• Diagonal 1 = 21 cm
• Diagonal 2 = 24 cm .
• Area of Rhombus = 252 cm²

Now ,

• $\implies{\sf{\large { 252 cm²\: =\: \frac {1}{2} \times 21 \times 24}}}$
• $\implies{\sf{\large { 252 cm²\: =\: 21 \times 12}}}$
• $\implies{\sf{\large { 252 cm²\: =\: 252cm²}}}$

Therefore,

• $\underline{\boxed{\star{\sf{\blue{ LHS = RHS.}}}}}$
• $\underline{\boxed{\star{\sf{\blue{ Hence\:Verified!.}}}}}$

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