The table at right shows measurements made on a circle with a one-meter radius. Each entry in the s-column is an arc length,and the adjacent entry in the c-column is the corresponding chord length, both in meters. Explain why c < s, and determine the range of values for c and for s. With s on the horizontal axis and c on the vertical axis, sketch an approximate graph of
c vs s.
s c
0.256 0.255
0.618 0.608
1.234 1.157
1.571 1.414
1.896 1.625
Then.(Continuation)Express s and c in terms of è, the central angle that intercepts s and c. Combine these equations to express c as a function of s. Graph this relationship.
idk this question either and the explanations above dont help
In a circle, a chord is a line segment that connects any two points on the circumference of the circle, while an arc is a portion of the circumference itself. In the given table, the s-column represents the arc length, and the c-column represents the corresponding chord length.
To understand why c < s, we need to consider the properties of a circle. We know that the chord length is always shorter than the corresponding arc length because the chord creates a straight line between two points on the circumference, while the arc follows along the circumference itself. This can be visually observed when considering the extreme cases of a very short chord (almost a straight line) compared to its corresponding arc.
Now, let's determine the range of values for c and s based on the given table:
The smallest values in the table are c = 0.255 and s = 0.256.
The largest values in the table are c = 1.625 and s = 1.896.
Therefore, the range of values for c is between 0.255 and 1.625, and the range of values for s is between 0.256 and 1.896.
To express s and c in terms of the central angle è that intercepts them, we can use the relationship between arc length and the central angle.
The formula to calculate the arc length is s = r * è, where r is the radius of the circle and è is the central angle in radians.
Similarly, we can use the law of sines to find the chord length. The formula is c = 2 * r * sin(è/2).
Combining these equations, we get:
c = 2 * r * sin(è/2)
Now, we can graph this relationship between c and s. Since c = 2 * r * sin(è/2), we can express c as a function of s using the equation:
c = 2 * r * sin(s/(2 * r))
Graphing this relationship would require specific values for the radius and measuring units, but in general, you would take the values of s along the horizontal axis and calculate the corresponding values of c using the equation above. Then, plot the points (s, c) on a graph to obtain the approximate graph of c vs s.
Well, it seems like you've got a case of the "circle blues"! Don't worry, I'm here to bring a little amusement to your geometry troubles.
Now, let's talk about why c is always less than s. You see, the chord length (c) is basically the straight line between two points on a circle, while the arc length (s) is the actual length of the curve along the circumference. Since the chord cuts straight through the circle, it's always going to be shorter than the arc length that curves around it.
As for the range of values for c and s, we can see from the measurements given that both c and s are always positive and increasing. Therefore, c and s can take on any positive real values.
Now, concerning the sketch of c vs s, let's imagine we're in the circus! Imagine c as a tightrope walker trying to keep up with s, the daredevil cyclist. As s increases, c will try to follow along beneath it, always slightly behind. So, our graph will show a line that starts near the origin and gradually climbs higher, just trailing behind the curve of s.
As for expressing c and s in terms of the central angle, è, well, that would involve some fancy math tricks that even the greatest clowns would find challenging. But hey, who needs complicated equations when we can rely on the reliable relationship of c being a little shorter than s?
Remember, a little laughter goes a long way, even when dealing with circles!
We can't draw graphs for you. You have the data to draw one yourself.
Is your symbol è supposed to be theta? Let's just call it angle A. I will assume it is in radians.
The arc length is
s = Radius*A = A
The chord length is
c = 2 R sin (A/2) = 2 sin (A/2)
c = 2 sin (s/2)
The data in the table fits that equation well.
USE THE GIVEN SET OF POINTS TO PROVE EACH CONGRUENCE STATEMENT
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