A chord of a circle 38.8 cm long is subtended by a 150 degree central angle.Calculate the radius of the circle.
sin 75° = 19.4/r
r = 20.08
Well, let's try to solve this problem without getting too tangled up!
First, we need to remember a little geometry. In a circle, the length of a chord is related to the radius by this formula: chord length = 2 * radius * sin(angle/2).
So, for this problem, the chord length is given as 38.8 cm and the central angle is 150 degrees.
Plugging those numbers into the formula, we get:
38.8 = 2 * radius * sin(150/2)
Now, let's simplify things a bit. The sine of 75 degrees is √3/2, so:
38.8 = 2 * radius * (√3/2)
To isolate the radius, we can divide both sides of the equation by 2 * (√3/2):
38.8 / (2 * (√3/2)) = radius
Well, that's a bit of a mouthful! So, let's see if we can simplify that expression a little.
Dividing 38.8 by 2 gives us 19.4, and dividing (√3/2) by itself is just 1. So:
19.4 / (√3/2) = radius
Now, let's rationalize the denominator by multiplying the top and bottom by the conjugate of (√3/2). That gives us:
(19.4 * 2) / (√3) = radius
Simplifying a little further:
38.8 / (√3) = radius
So, the radius of the circle is approximately 38.8 / (√3) cm.
To calculate the radius of the circle, we will use the formula:
\[ r = \frac{c}{2\sin(\frac{\theta}{2})} \]
where:
- \( c \) is the length of the chord
- \( \theta \) is the central angle in radians
First, we need to convert the central angle from degrees to radians:
\( \theta = \frac{150}{180}\pi \)
Now we can substitute the values into the formula:
\[ r = \frac{38.8}{2\sin(\frac{150}{180}\pi)} \]
Simplifying further:
\[ r = \frac{38.8}{2\sin(\frac{5\pi}{6})} \]
Using the sine of \( \frac{5\pi}{6} \) (which is \( \frac{1}{2} \)):
\[ r = \frac{38.8}{2\cdot \frac{1}{2}} \]
Final calculation:
\[ r = 38.8 \]
Therefore, the radius of the circle is 38.8 cm.
To solve this problem, we first need to understand some key concepts related to circles and chords.
1. In a circle, a chord is a straight line segment whose endpoints are on the circle.
2. The length of a chord can be determined using the formula: chord length = 2 * radius * sin(angle/2), where "angle" is the central angle subtended by the chord and "radius" is the radius of the circle.
Now, let's solve the problem step by step:
1. Given that the chord length (c) is 38.8 cm and the central angle (θ) is 150 degrees, we can plug these values into the formula:
c = 2 * radius * sin(θ/2)
2. Rearranging the formula to solve for the radius (r), we get:
radius (r) = c / (2 * sin(θ/2))
3. Plugging in the values, we have:
radius (r) = 38.8 cm / (2 * sin(150°/2))
4. Calculating the value inside the sin function:
θ/2 = 150°/2 = 75°
radius (r) = 38.8 cm / (2 * sin(75°))
5. Compute the sine of 75 degrees using a scientific calculator or online tool:
sin(75°) ≈ 0.9659
6. Substitute the value of sin(75°) into the formula:
radius (r) ≈ 38.8 cm / (2 * 0.9659)
7. Perform the calculations:
radius (r) ≈ 38.8 cm / 1.9318 ≈ 20.09 cm
Therefore, the radius of the circle is approximately 20.09 cm.