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Given: A right triangle is given, and an altitude is drawn to the hypotenuse of the triangle.
Required: To determine the missing side x.
Explanation: The given triangle is as follows-
Let the side of the triangle be as shown in the figure. Now triangle ABD is a right-angled triangle. Hence, by Pythagoras theorem, we have-
[tex]\begin{gathered} BD^2=AB^2+AD^2 \\ (225)^2=x^2+y^2\text{ ...}(1) \end{gathered}[/tex]Similarly, triangles ABC and ADC are right-angled triangles. Thus-
[tex]\begin{gathered} y^2=z^2+(144)^2\text{ ...}(2) \\ x^2=(81)^2+z^2\text{ }...(3) \end{gathered}[/tex]Equations (1), (2), and (3) represent equations in 3 variables. Hence solving equations (1) and (2) by substituting the value of y from equation (2) into equation (1) as follows-
[tex]\begin{gathered} x^2+z^2+(144)^2=(225)^2 \\ x^2+z^2=(225+144)(225-144) \\ x^2+z^2=369\times81 \\ x^2+z^2=29889\text{ ...}(4) \end{gathered}[/tex]Now, we can solve equations (3) and (4) for x as follows-
[tex]x^2+x^2+z^2=6561+z^2+29889[/tex]Further solving for x as-
[tex]\begin{gathered} 2x^2=36450 \\ x=\sqrt{18225} \\ x=\pm135\text{ units} \end{gathered}[/tex]Since the side of a triangle can't be negative. Hence, x=135 units.
Final Answer: The length of the missing side is-
[tex]x=135\text{ units}[/tex]