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 = 12, s_1 = 6 )
( s_2 = square )
( a = square )
( u_1 = square )
( u_2 = square )
(simplify your answers. type exact answers, using radicals as needed.)

Explanation:

Step1: Find $s_2$ via altitude rule

For a right triangle with altitude $h$ to hypotenuse, $h^2 = s_1s_2$.
$$12^2 = 6s_2 \implies 144 = 6s_2 \implies s_2 = \frac{144}{6} = 24$$

Step2: Find $u_1$ via leg rule

The leg $u_1$ satisfies $u_1^2 = s_1(s_1+s_2)$.
First calculate $s_1+s_2 = 6+24=30$.
$$u_1^2 = 6 \times 30 = 180 \implies u_1 = \sqrt{180} = 6\sqrt{5}$$

Step3: Find $u_2$ via leg rule

The leg $u_2$ satisfies $u_2^2 = s_2(s_1+s_2)$.
$$u_2^2 = 24 \times 30 = 720 \implies u_2 = \sqrt{720} = 12\sqrt{5}$$

Step4: Find $a$ (hypotenuse)

$a = s_1 + s_2$
$$a = 6 + 24 = 30$$

Answer:

$s_2 = 24$
$a = 30$
$u_1 = 6\sqrt{5}$
$u_2 = 12\sqrt{5}$