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 = 38 ), ( s_1 = 19 )
( s_2 = square )
( a = square )
( u_1 = square )
( u_2 = square )
(simplify your answers. type exact answers, using radicals as needed)

Explanation:

Step1: Find $s_2$ (geometric mean rule)

We use the property that $h^2 = s_1 \times s_2$. Rearrange to solve for $s_2$:
$$s_2 = \frac{h^2}{s_1} = \frac{38^2}{19} = \frac{1444}{19} = 76$$

Step2: Find $u_1$ (Pythagorean theorem)

Use right triangle with legs $s_1, h$:
$$u_1 = \sqrt{s_1^2 + h^2} = \sqrt{19^2 + 38^2} = \sqrt{361 + 1444} = \sqrt{1805} = 19\sqrt{5}$$

Step3: Find $u_2$ (Pythagorean theorem)

Use right triangle with legs $s_2, h$:
$$u_2 = \sqrt{s_2^2 + h^2} = \sqrt{76^2 + 38^2} = \sqrt{5776 + 1444} = \sqrt{7220} = 38\sqrt{5}$$

Step4: Find $a$ (geometric mean rule)

We use the property that $a^2 = s_1 \times s_2$ (or $a = \sqrt{s_1 \times s_2}$):
$$a = \sqrt{19 \times 76} = \sqrt{1444} = 38$$

Answer:

$s_2 = 76$
$a = 38$
$u_1 = 19\sqrt{5}$
$u_2 = 38\sqrt{5}$