Sum of the Squares of Adjacent Sides of a Parallelogram is 130 Sq.Cm and Length of One of Its Diagonals is 14 Cm. Find the Length of the Other Diagonal. - Geometry Mathematics 2

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Sum

Sum of the squares of adjacent sides of a parallelogram is 130 sq.cm and length of one of its diagonals is 14 cm. Find the length of the other diagonal.

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Solution

It is given that,
AB2 + AD2 = 130 sq. cm
BD = 14 cm

Diagonals of a parallelogram bisect each other.
i.e. O is the midpoint of AC and BD.

In ∆ABD, point O is the midpoint of side BD.

\[BO = OD = \frac{1}{2}BD = 7 cm\]

\[{AB}^2 + {AD}^2 = 2 {AO}^2 + 2 {BO}^2 \left( \text{by Apollonius theorem} \right)\]
\[ \Rightarrow 130 = 2 {AO}^2 + 2 \left( 7 \right)^2 \]
\[ \Rightarrow 130 = 2 {AO}^2 + 2 \times 49\]
\[ \Rightarrow 130 = 2 {AO}^2 + 98\]
\[ \Rightarrow 2 {AO}^2 = 130 - 98\]
\[ \Rightarrow 2 {AO}^2 = 32\]
\[ \Rightarrow {AO}^2 = 16\]
\[ \Rightarrow AO = 4 cm\]

Since point O is the midpoint of side AC.

∴ AC = 2AO = 8 cm

Hence, the length of the other diagonal is 8 cm.

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Chapter 2: Pythagoras Theorem - Problem Set 2 [Page 45]

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Balbharati Mathematics 2 Geometry 10th Standard SSC Maharashtra State Board
Chapter 2 Pythagoras Theorem
Problem Set 2 | Q 12 | Page 45

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