Hypotenuse, euclid

In a right-angled triangle, the hypotenuse has a length of 24 cm. The foot of the altitude to the hypotenuse divides it into two parts in a ratio of 2:4. What is the length of the altitude to the hypotenuse in cm? Calculate the perimeter of this right triangle in centimeters.

Final Answer:

h =  11.3137 cm
o =  57.4523 cm

Step-by-step explanation:

$$\begin{gathered} c=24 \\ c=c_1+c_2 \\ {c}_1:c_2=2:4 \\ {c}_1=c\cdot \frac{2}{2+4}=24\cdot \frac{2}{2+4}=8\text{cm} \\ {c}_2=c\cdot \frac{4}{2+4}=24\cdot \frac{4}{2+4}=16\text{cm} \\ h=\sqrt{c_1\cdot c_2}=\sqrt{8\cdot 16}=8\sqrt{2}=11.3137\text{cm} \end{gathered}$$

$$\begin{gathered} a=\sqrt{c_1\cdot c}=\sqrt{8\cdot 24}=8\sqrt{3}\text{cm}\doteq 13.8564\text{cm} \\ b=\sqrt{c_2\cdot c}=\sqrt{24\cdot 24}=8\sqrt{6}\text{cm}\doteq 19.5959\text{cm} \\ {c}_2=\sqrt{a^2+b^2}=\sqrt{13.8564^2+19.5959^2}=24\text{cm} \\ o=a+b+c=13.8564+19.5959+24=57.4523\text{cm} \end{gathered}$$

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