Triangle Euclid parameters

Using Euclid's Theorems and Pythagoras' Theorem, complete the following parameters describing a right-angled triangle ABC with a right angle at vertex C if we know b=10, cb=8

Final Answer:

a =  7.5
c =  12.5
c₁ =  4.5
S =  37.5
R =  6.25
h =  6

Step-by-step explanation:

$$\begin{gathered} b=10 \\ {c}_2=8 \\ c=c_1+c_2 \\ {c}^2=a^2+b^2 \\ {b}^2=c\cdot c_2 \\ c=b^2/c_2=10^2/8=\frac{25}{2}=12\frac{1}{2}=12.5 \\ a=\sqrt{c^2-b^2}=\sqrt{12.5^2-10^2}=7.5 \end{gathered}$$

$c_1=c-c_2=12.5-8=4.5$

$S=\frac{a\cdot b}{2}=\frac{7.5\cdot 10}{2}=37.5$

$R=c/2=12.5/2=6.25$

$$\begin{gathered} h=\sqrt{c_1\cdot c_2}=\sqrt{4.5\cdot 8}=6 \\ \text{ Verifying Solution: } \\ S=\frac{a\cdot b}{2}=\frac{c\cdot h}{2} \\ a\cdot b=c\cdot h \\ {h}_2=a\cdot b/c=7.5\cdot 10/12.5=6 \\ {h}_2=h \end{gathered}$$

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