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Question

# Adjacent sides of a parallelogram are 11 cm and 17 cm. If the length of one of its diagonals is 26 cm, find the length of the other diagonal.

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Solution

## First, we draw a figure of the parallelogram ABCD in which AB = 17 cm and BC = 11 cm, are the adjacent sides and AC and BD are the diagonals that bisect each other at O. Thus, AO = OC = $\frac{AC}{2}$ and BO = OD = $\frac{\mathrm{BD}}{2}$ …(1) In Δ ABC, BO is a median of the triangle. Thus, on using Appollonius' Theorem, we get: ${\mathrm{AB}}^{2}+{\mathrm{BC}}^{2}=2{\mathrm{BO}}^{2}+2{\mathrm{AO}}^{2}\phantom{\rule{0ex}{0ex}}{\mathrm{AB}}^{2}+{\mathrm{BC}}^{2}=2{\left(\frac{\mathrm{BD}}{2}\right)}^{2}+2{\left(\frac{\mathrm{AC}}{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}{\mathrm{AB}}^{2}+{\mathrm{BC}}^{2}=2×\frac{{\mathrm{BD}}^{2}}{4}+2×\frac{{\mathrm{AC}}^{2}}{4}\phantom{\rule{0ex}{0ex}}{\mathrm{AB}}^{2}+{\mathrm{BC}}^{2}=\frac{{\mathrm{BD}}^{2}}{2}+\frac{{\mathrm{AC}}^{2}}{2}\phantom{\rule{0ex}{0ex}}\mathrm{AB}=17\mathrm{cm},\mathrm{BC}=11\mathrm{cm}\text{and}\mathrm{AC}=26\mathrm{cm}\phantom{\rule{0ex}{0ex}}\text{So},\left(17{\right)}^{2}+\left(11{\right)}^{2}=\frac{{\mathrm{BD}}^{2}}{2}+\frac{{\left(26\right)}^{2}}{2}\phantom{\rule{0ex}{0ex}}⇒289+121=\frac{{\mathrm{BD}}^{2}}{2}+\frac{676}{2}\phantom{\rule{0ex}{0ex}}⇒410=\frac{{\mathrm{BD}}^{2}}{2}+338\phantom{\rule{0ex}{0ex}}⇒\frac{{\mathrm{BD}}^{2}}{2}=410-338\phantom{\rule{0ex}{0ex}}⇒{\mathrm{BD}}^{2}=2×72\phantom{\rule{0ex}{0ex}}⇒{\mathrm{BD}}^{2}=144\phantom{\rule{0ex}{0ex}}⇒\mathrm{BD}=\sqrt{144}=12\mathrm{cm}$

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