Question

an inscribed angle and a central angle intercept the same arc. the inscribed angle is ______ the arc.
select one:
a. half
b. twice
c. equal to

a tree casts a shadow 30 feet long when the angle of the sun is 32 degrees. how tall is the tree?
select one:
a. 18.7 feet
b. 56.6 feet
c. 25.4 feet
d. 15.9 feet

Explanation:

First Question

This relies on a core circle theorem: an inscribed angle intercepting an arc has a measure equal to half the measure of its intercepted arc, while a central angle is equal to its intercepted arc.

Step1: Define variables and trigonometric relation

Let $h$ = height of the tree. We use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{\text{shadow length}}$

Step2: Substitute known values

$\tan(32^\circ) = \frac{h}{30}$

Step3: Solve for $h$

$h = 30 \times \tan(32^\circ)$
Calculate $\tan(32^\circ) \approx 0.6249$, so $h \approx 30 \times 0.6249 = 18.747$

Answer:

a. half

---

Second Question