Triangle perimeter ratio

The perimeter of the ABC triangle is 19.6 cm. The following applies to the lengths of its sides: a: c = 1:2, b: c = 5:6. Calculate the lengths of all sides of the triangle ABC.

Final Answer:

a =  3.6296 cm
b =  7.2593 cm
c =  8.7111 cm

Step-by-step explanation:

$$\begin{gathered} o=19.6\text{cm} \\ a:c=1:2=5:10 \\ b:c=5:6=10:12 \\ a:b:c=5:10:12 \\ d=\frac{o}{5+10+12}=\frac{19.6}{5+10+12}=\frac{98}{135}\doteq 0.7259\text{cm} \\ a=5\cdot d=5\cdot \frac{98}{135}=\frac{5\cdot 98}{135}=\frac{490}{135}=\frac{98}{27}\text{cm}=3.6296\text{cm} \end{gathered}$$

$b=10\cdot d=10\cdot \frac{98}{135}=\frac{10\cdot 98}{135}=\frac{980}{135}=\frac{196}{27}\text{cm}=7.2593\text{cm}$

$$\begin{gathered} c=12\cdot d=12\cdot \frac{98}{135}=\frac{12\cdot 98}{135}=\frac{1176}{135}=\frac{392}{45}\doteq 8.7111\text{cm} \\ \text{ Verifying Solution: } \\ {r}_1=a/b=\frac{98}{27}/\frac{196}{27}=\frac{98}{27}:\frac{196}{27}=\frac{98}{27}\cdot \frac{27}{196}=\frac{98\cdot 27}{27\cdot 196}=\frac{2646}{5292}=\frac{1}{2}=0.5 \\ {r}_2=b/c=\frac{196}{27}/\frac{392}{45}=\frac{196}{27}:\frac{392}{45}=\frac{196}{27}\cdot \frac{45}{392}=\frac{196\cdot 45}{27\cdot 392}=\frac{8820}{10584}=\frac{5}{6}\doteq 0.8333 \\ {o}_2=a+b+c=\frac{98}{27}+\frac{196}{27}+\frac{392}{45}=\frac{490}{135}+\frac{980}{135}+\frac{1176}{135}=\frac{490+980+1176}{135}=\frac{2646}{135}=\frac{98}{5}=19.6\text{cm} \end{gathered}$$

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