Question

49 use trig ratios to find x. (triangle with x, 8, 24°) 50 what is the intercepted arc ab (circle with a, b, d, 42°) 51 what are both inscribed angles? (circle with x, y, 140°) bonus: what is the pythagorean theorem?

Explanation:

Problem 49: Use Trig Ratios to Find \( x \)

Step 1: Identify the trigonometric ratio

We have a right triangle with an angle of \( 24^\circ \), the opposite side to this angle is \( 8 \), and \( x \) is the hypotenuse. The sine ratio is defined as \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). So, \( \sin(24^\circ) = \frac{8}{x} \).

Step 2: Solve for \( x \)

Rearrange the formula to solve for \( x \): \( x = \frac{8}{\sin(24^\circ)} \). Calculate \( \sin(24^\circ) \approx 0.4067 \). Then \( x \approx \frac{8}{0.4067} \approx 19.67 \).

Step 1: Identify the central angle

In a circle, the measure of a central angle is equal to the measure of its intercepted arc. Here, \( \angle ADB = 42^\circ \) is a central angle intercepting arc \( \widehat{AB} \).

Step 2: Determine the arc measure

Since the central angle \( \angle ADB = 42^\circ \), the intercepted arc \( \widehat{AB} \) has the same measure as the central angle.

Step 1: Recall the inscribed angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. The arc given is \( 140^\circ \).

Step 2: Find \( x \) and \( y \)

Both \( x \) and \( y \) are inscribed angles intercepting the \( 140^\circ \) arc. So, \( x = \frac{1}{2} \times 140^\circ = 70^\circ \) and \( y = \frac{1}{2} \times 140^\circ = 70^\circ \) (since they intercept the same arc).

Answer:

\( x \approx 19.67 \) (or more precise value depending on calculator precision)

Problem 50: Intercepted Arc \( \widehat{AB} \)