Question

find the value of x using the figure to the right.
x = (simplify your answer.)

Explanation:

Step1: Apply the geometric mean theorem

In a right - triangle, if an altitude is drawn to the hypotenuse, then the length of the altitude is the geometric mean between the lengths of the segments of the hypotenuse. So, \((x + 15)^2=x\times60\).

Step2: Expand the left - hand side

Expand \((x + 15)^2\) using the formula \((a + b)^2=a^{2}+2ab + b^{2}\). Here \(a=x\) and \(b = 15\), so \((x + 15)^2=x^{2}+30x+225\). The equation becomes \(x^{2}+30x + 225=60x\).

Step3: Rearrange the equation to a quadratic form

Subtract \(60x\) from both sides: \(x^{2}+30x+225 - 60x=0\), which simplifies to \(x^{2}-30x + 225=0\).

Step4: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 1\), \(b=-30\), \(c = 225\)), we can use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) or factor. Factoring \(x^{2}-30x + 225\), we get \((x - 15)^2=0\) (since \((x-15)(x - 15)=x^{2}-30x + 225\)).

Answer:

\(15\)