Question

the diagram shows the parts of a right triangle with an altitude to the hypotenuse. for the two given measures, find the other four.
( h = 6, s_1 = 3 )
( s_2 = square )
( r = square )
( u_1 = square )
( u_2 = square )
(simplify your answers. type exact answers, using radicals as needed)

Explanation:

Step1: Find $u_1$ (altitude rule)

In a right triangle with altitude to hypotenuse, $h^2 = s_1 \cdot u_1$. Rearrange to solve for $u_1$:
$u_1 = \frac{h^2}{s_1} = \frac{6^2}{3} = \frac{36}{3} = 12$

Step2: Find $s_2$ (geometric mean)

First use $h^2 = s_1 \cdot s_2$ to solve for $s_2$:
$s_2 = \frac{h^2}{s_1} = \frac{6^2}{3} = 12$

Step3: Find $r$ (hypotenuse length)

Hypotenuse $r = s_1 + s_2 = 3 + 12 = 15$

Step4: Find $u_2$ (Pythagorean theorem)

Use $u_2^2 = s_2 \cdot r$:
$u_2 = \sqrt{s_2 \cdot r} = \sqrt{12 \cdot 15} = \sqrt{180} = 6\sqrt{5}$

Answer:

$s_2 = 12$
$r = 15$
$u_1 = 12$
$u_2 = 6\sqrt{5}$