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. u1 = 16, s2 = 24 s1 = (simplify your answers. type exact answers, using radicals as needed.) h = u2 = a =

Explanation:

Step1: Use geometric - mean theorem for $s_1$

In a right - triangle with altitude to the hypotenuse, $u_1^2=s_1\times h$ and also the geometric - mean theorem for segments of the hypotenuse gives $a^2 = s_1\times s_2$. Another important relation is $u_1^2=s_1\times(s_1 + s_2)$. Substituting $u_1 = 16$ and $s_2=24$, we have $16^2=s_1\times(s_1 + 24)$. Expanding gives $256=s_1^2+24s_1$. Rearranging to a quadratic equation $s_1^2+24s_1 - 256 = 0$. Using the quadratic formula $s_1=\frac{-24\pm\sqrt{24^2-4\times(- 256)}}{2}=\frac{-24\pm\sqrt{576 + 1024}}{2}=\frac{-24\pm\sqrt{1600}}{2}=\frac{-24\pm40}{2}$. We take the positive root, so $s_1=\frac{-24 + 40}{2}=8$.

Step2: Calculate $h$

Since $h=s_1 + s_2$, and $s_1 = 8$ and $s_2=24$, then $h=8 + 24=32$.

Step3: Calculate $u_2$

Using the geometric - mean theorem $u_2^2=s_2\times h$. Substituting $s_2 = 24$ and $h = 32$, we get $u_2^2=24\times32=768$. So $u_2=\sqrt{768}=16\sqrt{3}$.

Step4: Calculate $a$

Using the geometric - mean theorem $a^2=s_1\times s_2$. Substituting $s_1 = 8$ and $s_2=24$, we have $a^2=8\times24 = 192$. So $a=\sqrt{192}=8\sqrt{3}$.

Answer:

$s_1 = 8$, $h = 32$, $u_2=16\sqrt{3}$, $a = 8\sqrt{3}$