Triangle height ratio

In the right-angled triangle ABC (AB is the hypotenuse), a : b = 24 : 7, and the height to the side c = 12.6 cm applies. Calculate the lengths of the sides of triangle ABC.

Final Answer:

a =  45 cm
b =  13.125 cm
c =  46.875 cm

Step-by-step explanation:

$$\begin{gathered} a:b=24:7=24x:7x \\ h=12.6\text{cm} \\ {c}^2=a^2+b^2=(24x{)}^2+(7x{)}^2 \\ c=x\cdot \sqrt{24^2+7^2} \\ c=x\cdot 25=25x \\ S=\frac{a\cdot b}{2}=\frac{c\cdot h}{2} \\ a\cdot b=c\cdot h \\ 24x\cdot 7\cdot x=25\cdot x\cdot h \\ 24x\cdot 7=25\cdot h \\ x=\frac{25\cdot h}{24\cdot 7}=\frac{25\cdot 12.6}{24\cdot 7}=\frac{15}{8}=1.875\text{cm} \\ a=24\cdot x=24\cdot 1.875=45\text{cm} \end{gathered}$$

$b=7\cdot x=7\cdot 1.875=13.125\text{cm}$

$$\begin{gathered} c=25\cdot x=25\cdot 1.875=\frac{375}{8}=46.875=46.875\text{cm} \\ \text{ Verifying Solution: } \\ r=a/b=45/13.125=\frac{24}{7}\doteq 3.4286 \\ {c}_2=\sqrt{a^2+b^2}=\sqrt{45^2+13.125^2}=\frac{375}{8}=46.875\text{cm} \end{gathered}$$

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