Perimeter of a triangle

If the perimeter of a triangle is 6 2/3 cm and the lengths of two of its sides are 2 1/2 cm and 3 1/3 cm, find the length if the third side

Correct answer:

c =  5/6 cm

Step-by-step explanation:

$$\begin{gathered} p=6\frac{2}{3}=6+\frac{2}{3}=\frac{6\cdot 3+2}{3}=\frac{20}{3}\doteq 6.6667\text{cm} \\ a=2\frac{1}{2}=2+\frac{1}{2}=\frac{2\cdot 2+1}{2}=\frac{5}{2}=2.5\text{cm} \\ b=3\frac{1}{3}=3+\frac{1}{3}=\frac{3\cdot 3+1}{3}=\frac{10}{3}\doteq 3.3333\text{cm} \\ a+b+c=p \\ c=p-(a+b)=6.6667-(2.5+3.3333)=\frac{5}{6}\doteq 0.8333\text{cm} \\ \text{ Verifying Solution: } \\ {c}_1=a+b>c \\ {a}_1=b+c>a \\ {b}_1=c+a>b \\ {c}_1=a+b=2.5+3.3333=\frac{35}{6}\doteq 5.8333\text{cm} \\ {a}_1=b+c=3.3333+0.8333=\frac{25}{6}\doteq 4.1667\text{cm} \\ {b}_1=a_1+c_1=4.1667+5.8333=10\text{cm} \end{gathered}$$

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