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 = 30, s_1 = 15 )
( 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 geometric mean

In a right triangle, the altitude $h$ to the hypotenuse satisfies $h^2 = s_1s_2$.
$$30^2 = 15s_2 \implies 900 = 15s_2 \implies s_2 = \frac{900}{15} = 60$$

Step2: Find $u_1$ via Pythagorean theorem

$u_1$ is the hypotenuse of the small right triangle with legs $h$ and $s_1$.
$$u_1 = \sqrt{s_1^2 + h^2} = \sqrt{15^2 + 30^2} = \sqrt{225 + 900} = \sqrt{1125} = 15\sqrt{5}$$

Step3: Find $u_2$ via Pythagorean theorem

$u_2$ is the hypotenuse of the small right triangle with legs $h$ and $s_2$.
$$u_2 = \sqrt{s_2^2 + h^2} = \sqrt{60^2 + 30^2} = \sqrt{3600 + 900} = \sqrt{4500} = 30\sqrt{5}$$

Step4: Find $a$ (hypotenuse of big triangle)

$a = s_1 + s_2$
$$a = 15 + 60 = 75$$

Answer:

$s_2 = 60$
$a = 75$
$u_1 = 15\sqrt{5}$
$u_2 = 30\sqrt{5}$