Answer:
[tex]\begin{gathered} x=\sqrt[]{35}=5.62 \\ y=\sqrt[]{60}=7.75 \\ z=\sqrt[]{84}=9.17 \end{gathered}[/tex]
Explanation: We need to find x y z, three missing sides:
From the given triangle, we can form three equations, using the Pythagorean theorem as:
[tex]\begin{gathered} (1)\rightarrow5^2+x^2=y^2 \\ (2)\rightarrow7^2+x^2=z^2 \\ (3)\rightarrow z^2+y^2=(7+5)^2=12^2 \\ \end{gathered}[/tex]
Solution by substitution:
Substituting (1) in (3) gives:
[tex]\begin{gathered} z^2+5^2+x^2=12^2 \\ z^2+25+x^2=144 \\ z^2+x^2=144-25=119 \\ z^2+x^2=119\rightarrow(4) \end{gathered}[/tex]
We have reached equation (4), solving for z in equation (4), and then substituting it into equation (2) gives:
[tex]\begin{gathered} 7^2+x^2=119-x^2\rightarrow2x^2=119-49=70 \\ \therefore\rightarrow \\ x=\sqrt[]{\frac{70}{2}}=\sqrt[]{35}=5.916 \end{gathered}[/tex]
Plugging this x into equation (4) gives:
[tex]\begin{gathered} z^2+(\sqrt[]{35})^2=119\rightarrow z^2=119-35=84 \\ \therefore\rightarrow \\ z=\sqrt[]{84}=9.165 \\ \\ \end{gathered}[/tex]
Now that we have x and z, we plug x it into equation (1) and we get:
[tex]\begin{gathered} 5^2+(\sqrt[]{35})^2=y^2 \\ \therefore\rightarrow \\ y^2=25+35=60 \\ y=\sqrt[]{60}=7.75 \end{gathered}[/tex]
x y z respectively are:
[tex]\begin{gathered} x=\sqrt[]{35}=5.62 \\ y=\sqrt[]{60}=7.75 \\ z=\sqrt[]{84}=9.17 \end{gathered}[/tex]
