In a circle O draw a chord AB such that the length of major arc AB is twice the length of minor arc AB. How does the length of chord AB relate to the radius of the circle? I don’t get it.
Try sketching it out first : )
To understand the relationship between the length of the chord AB and the radius of the circle, let's break down the problem step by step.
1. Start by drawing a circle with center O.
2. Draw a chord AB inside the circle. A chord is a line segment that connects two points on a circle.
3. Let's assume the length of the minor arc AB is x units. Therefore, the length of the major arc AB is twice that, i.e., 2x units.
4. Now, let's examine the central angles corresponding to the arcs AB. By definition, the central angle that intercepts minor arc AB is equal to the central angle intercepting the major arc AB.
5. Since the length of the minor arc AB is x units, the central angle that intercepts this arc will be x degrees.
6. Similarly, the length of the major arc AB being 2x units implies that the central angle intercepting it will be 2x degrees.
7. The next key observation is that the central angle is directly proportional to the length of the arc it intercepts. Therefore, we can express this proportionality as:
x degrees ↔ x units
2x degrees ↔ 2x units
8. Considering that the entire circle has a central angle of 360 degrees, we can write:
x degrees + 2x degrees = 360 degrees
9. Simplifying the equation, we get:
3x degrees = 360 degrees
10. Solving for x, we find:
x = 120 degrees
11. Now, let's consider the triangle AOB formed by connecting points A, O, and B.
12. In a circle, the radius is perpendicular to the tangent at the point of contact. Therefore, segment OA and segment OB are radii of the circle.
13. In triangle AOB, the central angle at O is 2x degrees. Since the sum of the angles in a triangle is 180 degrees, the other two angles are (180 - 2x) degrees.
14. Considering that angle AOB subtends arc AB, which has a length of 2x units, we have the following relationship:
(180 - 2x) degrees ↔ 2x units
15. Now, let's use the relationship between angles and arc lengths to solve for the length of the chord AB.
16. In general, the length of an arc is given by the formula:
Length of arc = (angle at center ÷ 360) × circumference of circle
17. In our case, the angle at the center is 2x degrees, and the circumference of the circle is 2πr, where r is the radius of the circle.
Length of arc AB = (2x ÷ 360) × 2πr
18. Simplifying the equation, we get:
2x units = (x ÷ 180) × 2πr
19. Canceling out like terms, we find:
2 = (x ÷ 180) × πr
20. Solving for x, we substitute x = 120 degrees:
2 = (120 ÷ 180) × πr
2 = (2/3)πr
21. Finally, rearranging the equation, we have:
r = (3/2) units
Therefore, the length of the chord AB is related to the radius of the circle by the equation r = (3/2) times the length of the chord AB.