Question

the given diagram shows the parts of a right triangle with an altitude to the hypotenuse. using the two given measures, find the other four. s1 = 8, u2 = 3√17 s2 = (simplify your answers. type exact answers, using radicals as needed.) h = u1 = a =

Explanation:

Step1: Use geometric - mean theorem

In a right - triangle with altitude \(a\) to the hypotenuse \(h = s_1 + s_2\), we have the following relationships: \(a^{2}=s_1\times s_2\) and \(u_1^{2}=s_1\times h\), \(u_2^{2}=s_2\times h\). First, from the proportion \(\frac{s_1}{a}=\frac{a}{s_2}\), we know that \(a^{2}=s_1\times s_2\). Also, from similar triangles, \(\frac{u_1}{h}=\frac{s_1}{u_1}\) and \(\frac{u_2}{h}=\frac{s_2}{u_2}\).
We know that \(\frac{s_1}{u_1}=\frac{u_1}{s_1 + s_2}\) and \(\frac{s_2}{u_2}=\frac{u_2}{s_1 + s_2}\). Using the property \(u_2^{2}=s_2\times(s_1 + s_2)\). Given \(s_1 = 8\) and \(u_2=3\sqrt{17}\), then \((3\sqrt{17})^{2}=s_2\times(8 + s_2)\).

Step2: Solve the quadratic equation

Expand \((3\sqrt{17})^{2}=s_2\times(8 + s_2)\) to get \(153=8s_2+s_2^{2}\). Rearrange it to the standard quadratic - form \(s_2^{2}+8s_2 - 153 = 0\).
Using the quadratic formula \(s_2=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for the quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b = 8\), \(c=-153\)), we have \(s_2=\frac{-8\pm\sqrt{8^{2}-4\times1\times(-153)}}{2\times1}=\frac{-8\pm\sqrt{64 + 612}}{2}=\frac{-8\pm\sqrt{676}}{2}=\frac{-8\pm26}{2}\). We take the positive root \(s_2=\frac{-8 + 26}{2}=9\) (since length cannot be negative).

Step3: Calculate the altitude \(a\)

Since \(a^{2}=s_1\times s_2\) and \(s_1 = 8\), \(s_2 = 9\), then \(a=\sqrt{s_1\times s_2}=\sqrt{8\times9}=\sqrt{72}=6\sqrt{2}\).

Step4: Calculate \(u_1\)

Using the proportion \(\frac{s_1}{u_1}=\frac{u_1}{s_1 + s_2}\), or \(u_1^{2}=s_1\times(s_1 + s_2)\). Substitute \(s_1 = 8\) and \(s_2 = 9\), then \(u_1^{2}=8\times(8 + 9)=8\times17\), so \(u_1=\sqrt{136}=2\sqrt{34}\).

Step5: Calculate the hypotenuse \(h\)

\(h=s_1 + s_2=8 + 9 = 17\).

Answer:

\(s_2 = 9\), \(h = 17\), \(u_1=2\sqrt{34}\), \(a = 6\sqrt{2}\)