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Question

# 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 mid point of AC and BD. In ∆ABD, point O is the midpoint of side BD. $\mathrm{BO}=\mathrm{OD}=\frac{1}{2}\mathrm{BD}=7\mathrm{cm}$ ${\mathrm{AB}}^{2}+{\mathrm{AD}}^{2}=2{\mathrm{AO}}^{2}+2{\mathrm{BO}}^{2}\left(\mathrm{by}\mathrm{Apollonius}\mathrm{theorem}\right)\phantom{\rule{0ex}{0ex}}⇒130=2{\mathrm{AO}}^{2}+2{\left(7\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒130=2{\mathrm{AO}}^{2}+2×49\phantom{\rule{0ex}{0ex}}⇒130=2{\mathrm{AO}}^{2}+98\phantom{\rule{0ex}{0ex}}⇒2{\mathrm{AO}}^{2}=130-98\phantom{\rule{0ex}{0ex}}⇒2{\mathrm{AO}}^{2}=32\phantom{\rule{0ex}{0ex}}⇒{\mathrm{AO}}^{2}=16\phantom{\rule{0ex}{0ex}}⇒\mathrm{AO}=4\mathrm{cm}$ Sinec, point O is the midpoint of side AC. $\therefore \mathrm{AC}=2\mathrm{AO}=8\mathrm{cm}$ Hence, the length of the other diagonal is 8 cm.

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