Question

calculator what is the value of x? round to the nearest tenth, if necessary. x = 4 x = 5 x = 6.2 x = 9.4

Explanation:

Step1: Recall geometric mean theorem

In a right triangle, the length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments. Also, each leg of the right triangle is the geometric mean of the hypotenuse and the segment adjacent to that leg. Here, triangle \(ABD\) is right - angled at \(A\), and \(AC\) is the altitude to hypotenuse \(BD\). So, for leg \(AB\) (length \(9\)) and segment \(x\) (adjacent to \(AB\)) and hypotenuse \(BD = 13\), we have the formula \(AB^{2}=x\times BD\)? Wait, no, the correct formula is that in a right triangle, if we have a right triangle with hypotenuse \(c\), and a segment of the hypotenuse \(a\) adjacent to leg \(l\), then \(l^{2}=a\times c\). Wait, actually, the geometric mean theorem (also known as the altitude - on - hypotenuse theorem) states that in a right triangle, the square of a leg is equal to the product of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So, in \(\triangle ABD\) with right angle at \(A\) and \(AC\perp BD\), we have \(AB^{2}=BC\times BD\). Let \(BC = x\), \(AB = 9\), \(BD=13\).

Step2: Solve for \(x\)

From the geometric mean theorem, we have the equation \(9^{2}=x\times13\). So, \(81 = 13x\). Then, to solve for \(x\), we divide both sides of the equation by \(13\): \(x=\frac{81}{13}\approx6.2\) (rounded to the nearest tenth).

Answer:

\(x = 6.2\)