A triangle has two sides that measure 2.5 cm and 16.5 cm. Which could be the measure of the third side?

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Answer 1 · 2016-12-08T07:28:38

Third side will have a value between $14$ $14$ and $19$ $19$ .

As the two sides measure $2.5$ $2.5$ $c m .$ $cm.$ and $16.5$ $16.5$ $c m .$ $cm.$ , let the included angle between them be $θ$ $theta$ .

It is evident that $0^{o} < θ < 180^{o}$ $0^o < theta < 180^o$ .

Then using the cosine formula of solving triangles,

the square on the third side will be given by

${2.5}^{2} + {16.5}^{2} - 2 \times 2.5 \times 16.5 \times cos θ$ $2.5^2+16.5^2-2xx2.5xx16.5xxcostheta$

$= 6.25 + 272.25 - 82.5 cos θ$ $=6.25+272.25-82.5costheta$

$= 278.5 - 82.5 cos θ$ $=278.5-82.5costheta$

and as $0^{o} < θ < 180^{o}$ $0^o < theta < 180^o$ , $- 1 < cos θ < 1$ $-1 < costheta < 1$

Hence the third side say $x$ $x$ will be given by

$\sqrt{278.5 - 82.5} < x < \sqrt{278.5 + 82.5}$ $sqrt(278.5-82.5) < x < sqrt(278.5+82.5)$

or $\sqrt{196} < x < \sqrt{361}$ $sqrt196< x< sqrt361$

or $14 < x < 19$ $14 < x <19$ .

Hence third side will have a value between $14$ $14$ and $19$ $19$ .