In the image, point A marks the center of the circle. Which two lengths must form a ratio of 1:2?
Diagram shows a circle centered at A. Diameters EF and GI is drawn. Radius AD is drawn between points G and F. Chord BC is drawn that intersects diameter EF.
A.
AF:AE
B.
BC:EF
C.
EF:AI
D.
AI:EF
E.
AI:AD
The correct answer is A. AF:AE.
We can see from the diagram that AF and AE are both radii of the circle with center A. Therefore, they are equal in length.
Since radii of a circle are always equal in length, the ratio of AF to AE will always be 1:1 or simply 1.
None of the other options (BC:EF, EF:AI, AI:EF, AI:AD) involve lengths that are necessarily equal to each other, so they cannot have a ratio of 1:2.
To find the two lengths that must form a ratio of 1:2, we need to analyze the given information in the diagram.
First, let's identify the relevant lengths in the diagram:
- AF: The length of the segment AF.
- AE: The length of the segment AE.
- BC: The length of the segment BC.
- EF: The length of the segment EF.
- AI: The length of the segment AI.
- AD: The length of the segment AD.
Now, let's analyze the given information:
- Point A is marked as the center of the circle.
- Diameters EF and GI are drawn.
- Since a diameter passes through the center of a circle, we can deduce that EI is also a diameter.
- The radius AD is drawn between points G and F.
- Since AO is a radius (OA = OD), we know that OI and OD are also radii.
- The chord BC intersects diameter EF.
- The chord is a segment that connects two points on the circumference of a circle.
By analyzing the information, we can conclude that the correct lengths that must form a ratio of 1:2 are:
C. EF:AI
Explanation:
- The segment EF is a diameter, which means it passes through the center A.
- The segment AI is a radius, which connects point A to point I.
- Since the radius AI connects the center A to a point on the circumference (point I), the length of AI is half the length of the diameter EF. Thus, the ratio of EF to AI is 1:2.
Therefore, the answer is C. EF:AI.
To determine which two lengths must form a ratio of 1:2, we need to analyze the given information in the diagram.
Given that point A marks the center of the circle, and a radius (AD) is drawn between points G and F, we can consider the following lengths:
1. AF and AE: These are the radii drawn from the center of the circle to points F and E, respectively. Since both lengths start from the center and extend to the same point on the circumference, they have the same length. Therefore, the ratio between AF and AE is always 1:1.
2. BC and EF: These are segments within the circle. The chord BC intersects the diameter EF. To determine if the ratio is 1:2, we need more information about the specific lengths of BC and EF. Without additional information, we cannot determine the ratio between BC and EF.
3. EF and AI: These are lengths within the circle. EF is the diameter, while AI is a radius drawn from the center to point I. The ratio between EF and AI is not necessarily 1:2, as it depends on the specific lengths.
4. AI and AD: These are lengths within the circle. AI is a radius drawn from the center to point I, while AD is a radius drawn from the center to point D. The ratio between AI and AD is always 1:1, as both lengths are radii that start from the center of the circle.
Based on the given information, the correct answer is A. AF:AE, as radii drawn from the center to any point on the circumference of a circle have the same length, resulting in a ratio of 1:1.