Question

The length of the diagonals of a rhombus are in the ratio 3 : 4 . If its perimeter is 80 CM then find the length of its side and the diagonals.

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Answer 1

GIVEN :-

• Ratio of Length Of Diagonals of rhombus = 3:4
• Perimeter of rhombus = 80 cm.

TO FIND :-

• The side of rhombus.
• The Diagonals of rhombus.

SOLUTION :-

$\setlength{ \unitlength}{20} \begin{picture}(6,6) \linethickness{1} \put(1,4){\line(1,0){6}} \put(4,4){\line(0,1){3}} \put(4,4){\line(0, - 1){3}}\qbezier(1,4)(4,1)(4,1)\qbezier(4,1)(7,4)(7,4)\qbezier(7,4)(7,4)(4,7)\qbezier(4,7)(4,7)(1,4)\put(0.8,3.5){ \bf{ A } }\put(4,0.5){ \bf{ B} }\put(7,4){ \bf{ C} }\put(4,7){ \bf{ D } }\put(4.2,3.5){ \bf{ O } }\end{picture}$

☛ Let the ratio constant be x.

➣ 1st Diagonal (D1) , AC = 3x

➣2nd Diagonal (D2) , BD = 4x

☯ ACCORDING TO QUESTION,

$\mapsto \boxed{\red{\bf Perimeter (Rhombus) = 4 \times side}} \\ \\ \mapsto \sf 80 cm = 4 \times side \\ \\ \mapsto \sf side = \dfrac{80}{4} \\ \\ \mapsto \boxed{ \blue{\sf side = 20 cm }}$

Hence the side of rhombus is 20 cm.

☛ As we know that diagonals of a rhombus bisect each other at 90°.

$\boxed{ \red{\sf \therefore AC \perp BD}} \\ \\ \mapsto \sf OA = OC = \dfrac{AC}{2} \\ \\ \mapsto \sf OA = OC=\underline{ \boxed{ \blue{\dfrac{3x}{2}}}} \\ \\ \mapsto \sf \: OB = OD = \dfrac{BD}{2} = \dfrac{4\cancel x}{2} \\ \\ \mapsto \sf OB = OD=\underline{ \boxed{ \blue { \: \bf \: 2x}}}$

☯ BY PYTHAGORAS THEOREM IN △ AOD,

$\mapsto \boxed{\red{\sf OA^{2} + OD^{2} = AD^{2}}} \\ \\ \mapsto \sf \bigg(\dfrac{3x}{2} \bigg)^{2} + (2x)^{2} = (20)^{2} \\ \\ \mapsto \sf \dfrac{9x^{2}}{4} + 4x^{2} = 400 \\ \\ \mapsto \sf \dfrac{9x^{2} + 16x^{2}}{4} = 400 \\ \\ \mapsto \sf \dfrac{25 x^{2}}{4} = 400 \\ \\ \mapsto \sf 25x^{2} = 400 \times 4 \\ \\ \mapsto \sf 25x^{2} = 1600 \\ \\ \mapsto \sf x^{2} = \dfrac{1600}{25} \\ \\ \mapsto \sf x^{2} = 64 \\ \\ \mapsto \sf x = \sqrt{64} \\ \\ \mapsto \boxed{\blue{\sf x = 8}}$

◉ Hence the value of x is 8.

➣ 1st Diagonal (D1) , AC = 3x = 3 × 8 = 24 cm.

➣2nd Diagonal (D2) , BD = 4x = 4 × 8 = 32 cm.

ADDITIONAL INFORMATION :-

☛ All the side of rhombus are congruent.

☛ The diagonals of a Rhombus bisect each other at 90°.

☛ Area of Rhombus = 1/2 × (D1 × D2) or Base × Height.

Answer 2

Step-by-step explanation:

solution is provided in the above pic

hope it helps you

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