Certainly! Let's solve the problem step-by-step.
We are given:
1. The radius of the circle, [tex]\( r = 3 \)[/tex] cm
2. The angle [tex]\( \angle OAB = 70^\circ \)[/tex]
Step 1: Converting the angle from degrees to radians
First, we need to convert the angle [tex]\(\angle OAB\)[/tex] from degrees to radians. The formula to convert degrees to radians is:
[tex]\[ \text{angle in radians} = \text{angle in degrees} \times \left(\frac{\pi}{180}\right) \][/tex]
So,
[tex]\[ \angle OAB_\text{rad} = 70^\circ \times \left(\frac{\pi}{180}\right) \approx 1.2217 \text{ radians} \][/tex]
Step 2: Calculate the length [tex]\( AB \)[/tex]
To find the length of the chord [tex]\( AB \)[/tex], we will use the formula:
[tex]\[ AB = 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right) \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle, and [tex]\( \theta \)[/tex] is the angle in radians.
Plugging in the given values,
[tex]\[ AB = 2 \cdot 3 \cdot \sin\left(\frac{1.2217}{2}\right) \][/tex]
[tex]\[ AB \approx 2 \cdot 3 \cdot \sin(0.61085) \][/tex]
[tex]\[ AB \approx 2 \cdot 3 \cdot 0.5736 \][/tex]
[tex]\[ AB \approx 3.4415 \text{ cm} \][/tex]
Step 3: Calculate the distance from the center [tex]\( O \)[/tex] to the chord [tex]\( AB \)[/tex]
To find the perpendicular distance from the center [tex]\( O \)[/tex] to the chord [tex]\( AB \)[/tex], we use the formula:
[tex]\[ d = r \cdot \cos\left(\frac{\theta}{2}\right) \][/tex]
Substituting the known values,
[tex]\[ d = 3 \cdot \cos\left(\frac{1.2217}{2}\right) \][/tex]
[tex]\[ d \approx 3 \cdot \cos(0.61085) \][/tex]
[tex]\[ d \approx 3 \cdot 0.8192 \][/tex]
[tex]\[ d \approx 2.4575 \text{ cm} \][/tex]
So, summarizing the results:
- The length of [tex]\( AB \)[/tex] is approximately [tex]\( 3.4415 \)[/tex] cm
- The distance from the center [tex]\( O \)[/tex] to the chord [tex]\( AB \)[/tex] is approximately [tex]\( 2.4575 \)[/tex] cm
