Triangle and its heights

Calculate the length of the sides of the triangle ABC, if v_a=5 cm, v_b=7 cm and side b is 5 cm shorter than side a.

Correct answer:

a =  17.5 cm
b =  12.5 cm
c =  7.84 cm

Step-by-step explanation:

$$\begin{gathered} \sin \gamma ={\displaystyle \frac{v_a}{b}}={\displaystyle \frac{v_b}{a}} \\ b=a-5 \\ a={\displaystyle \frac{5\cdot 7}{7-5}}=17.5\text{ cm} \end{gathered}$$

$b=a-5=12.5\text{ cm}$

$$\begin{gathered} \cos \gamma ={\displaystyle \frac{\sqrt{a^2-v_b^2}}{a}} \\ c=\sqrt{a^2+b^2-2ab\cos \gamma }=7.84\text{ cm} \end{gathered}$$

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