Question

Instructions: Find the missing length indicated.x=Check81225X

Answer 1

Given: A right triangle is given, and an altitude is drawn to the hypotenuse of the triangle.

Required: To determine the missing side x.

Step-by-step 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-

`\begin{gathered} BD^2=AB^2+AD^2 \\ (225)^2=x^2+y^2\text{ ...}(1) \end{gathered}`

Similarly, triangles ABC and ADC are right-angled triangles. Thus-

`\begin{gathered} y^2=z^2+(144)^2\text{ ...}(2) \\ x^2=(81)^2+z^2\text{ }...(3) \end{gathered}`

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-

`\begin{gathered} x^2+z^2+(144)^2=(225)^2 \\ x^2+z^2=(225+144)(225-144) \\ x^2+z^2=369*81 \\ x^2+z^2=29889\text{ ...}(4) \end{gathered}`

Now, we can solve equations (3) and (4) for x as follows-

`x^2+x^2+z^2=6561+z^2+29889`

Further solving for x as-

`\begin{gathered} 2x^2=36450 \\ x=√(18225) \\ x=\pm135\text{ units} \end{gathered}`

Since the side of a triangle can't be negative. Hence, x=135 units.

Final Answer: The length of the missing side is-

`x=135\text{ units}`