The perimeter of a triangle is 6p² − 4p + 9 and two of its sides are p² − 2p + 1 and 3p² − 5p + 3. Find the third side of the triangle.

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Solution

Let $a, b a n d c$ be the three sides of the triangle.

∴ Perimeter of the triangle = $(a + b + c)$

Given perimeter of the triangle = $6 p^{2} - 4 p + 9$
One side ( $a$ ) = $p^{2} - 2 p + 1$
Other side ( $b$ ) = $3 p^{2} - 5 p + 3$
Perimeter = $(a + b + c)$
$(6 p^{2} - 4 p + 9) = (p^{2} - 2 p + 1) + (3 p^{2} - 5 p + 3) + c 6 p^{2} - 4 p + 9 - p^{2} + 2 p - 1 - 3 p^{2} + 5 p - 3 = c (6 p^{2} - p^{2} - 3 p^{2}) + (- 4 p + 2 p + 5 p) + (9 - 1 - 3) = c 2 p^{2} + 3 p + 5 = c$

Thus, the third side is $2 p^{2} + 3 p + 5$ .

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