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.
$u_{1}=18,s_{2}=15$
$s_{1}=\square$
$h=\square$
$u_{2}=\square$
$a=\square$
(simplify your answers. type exact answers, using radicals as needed.)

Explanation:

Step1: Use geometric mean theorem for \(s_1\)

In a right - triangle with an altitude to the hypotenuse, \(s_1=\sqrt{u_1h}\). First, use the geometric mean theorem \(s_1s_2 = u_1u_2\) and \(a^{2}=s_1s_2\), \(u_1^{2}=s_1h\), \(u_2^{2}=s_2h\). Also, from the proportion \(\frac{u_1}{s_1}=\frac{s_2}{u_2}\), cross - multiply to get \(u_1u_2=s_1s_2\). Another important formula is \(u_1^{2}=s_1h\). Since \(u_1 = 18\) and \(s_2=15\), and using the formula \(s_1s_2=u_1u_2\) and \(u_1^{2}=s_1h\), \(u_2^{2}=s_2h\), \(a^{2}=s_1s_2\). First, from \(u_1^{2}=s_1h\) and \(u_2^{2}=s_2h\), we can also use the fact that \(\frac{u_1}{s_1}=\frac{s_2}{u_2}\) (similar triangles). Cross - multiplying gives \(u_1u_2=s_1s_2\). Also, \(u_1^{2}=s_1h\) implies \(h=\frac{u_1^{2}}{s_1}\), and \(u_2^{2}=s_2h\) implies \(h = \frac{u_2^{2}}{s_2}\). But we use the formula \(s_1=\frac{u_1^{2}}{s_2}\) (derived from \(u_1^{2}=s_1h\) and \(u_2^{2}=s_2h\) and \(s_1s_2 = u_1u_2\) and the similarity of triangles). Substitute \(u_1 = 18\) and \(s_2=15\) into \(s_1=\frac{u_1^{2}}{s_2}\), so \(s_1=\frac{18^{2}}{15}=\frac{324}{15}=\frac{108}{5} = 21.6\).

Step2: Calculate \(h\)

Use \(h=s_1 + s_2\). Substitute \(s_1=\frac{108}{5}\) and \(s_2 = 15=\frac{75}{5}\), then \(h=\frac{108 + 75}{5}=\frac{183}{5}=36.6\).

Step3: Calculate \(u_2\)

Use \(u_2=\sqrt{s_2h}\). Substitute \(s_2 = 15\) and \(h=\frac{183}{5}\), \(u_2=\sqrt{15\times\frac{183}{5}}=\sqrt{3\times183}=\sqrt{549}=3\sqrt{61}\approx23.43\).

Step4: Calculate \(a\)

Use \(a=\sqrt{s_1s_2}\). Substitute \(s_1=\frac{108}{5}\) and \(s_2 = 15\), \(a=\sqrt{\frac{108}{5}\times15}=\sqrt{108\times3}=\sqrt{324}=18\).

Answer:

\(s_1=\frac{108}{5}\), \(h=\frac{183}{5}\), \(u_2 = 3\sqrt{61}\), \(a = 18\)