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 = 40, s₁ = 20
s₂ =
a =
u₁ =
u₂ =
(simplify your answers. type exact answers, using radicals as needed)

Explanation:

Step1: Find $s_2$ via altitude rule

The altitude to the hypotenuse satisfies $h^2 = s_1s_2$. Rearrange to solve for $s_2$:
$$s_2 = \frac{h^2}{s_1} = \frac{40^2}{20} = 80$$

Step2: Find $u_1$ via leg rule

Leg $u_1$ satisfies $u_1^2 = s_1h$. Calculate $u_1$:
$$u_1 = \sqrt{s_1h} = \sqrt{20 \times 40} = \sqrt{800} = 20\sqrt{2}$$

Step3: Find $u_2$ via leg rule

Leg $u_2$ satisfies $u_2^2 = s_2h$. Calculate $u_2$:
$$u_2 = \sqrt{s_2h} = \sqrt{80 \times 40} = \sqrt{3200} = 40\sqrt{2}$$

Step4: Find $a$ via Pythagoras on small triangle

Use right triangle with legs $s_1, a$ and hypotenuse $u_1$:
$$a = \sqrt{u_1^2 - s_1^2} = \sqrt{(20\sqrt{2})^2 - 20^2} = \sqrt{800 - 400} = \sqrt{400} = 20$$

Answer:

$s_2 = 80$
$a = 20$
$u_1 = 20\sqrt{2}$
$u_2 = 40\sqrt{2}$