Question

in the image, point a marks the center of the circle. which two lengths must form a ratio of 1:2?
a. af:ae
b. bc:ee
c. ef:ai
d. ai:ef
e. ai:ad

Explanation:

Step1: Recall circle properties

In a circle, the radius is half the diameter. \( AI \), \( AF \), \( AD \), \( AE \) are radii (distance from center \( A \) to circumference), and \( EF \) (if \( EF \) is a chord related to diameter or radius? Wait, wait, let's check options. Wait, \( AI \) is radius, \( EF \): Wait, no, let's re-examine. Wait, \( A \) is center, so \( AI \), \( AF \), \( AD \), \( AE \) are radii (length \( r \)), and if \( EF \) is a chord? Wait no, wait the options: Option D is \( AI:EF \)? Wait no, wait maybe \( EF \) is a diameter? Wait no, \( AI \) is radius. Wait, no, let's think again. Wait, in a circle, radius is half the diameter. So if \( AI \) is radius, and \( EF \) is diameter? Wait no, maybe I misread. Wait, the options: Let's check each option.

Option A: \( AF/AE \): Both \( AF \) and \( AE \) are radii, so ratio \( 1:1 \), not \( 1:2 \).

Option B: \( BC/EF \): \( BC \) and \( EF \) are chords, not sure, ratio not necessarily \( 1:2 \).

Option C: \( EF/AI \): If \( EF \) is diameter (length \( 2r \)) and \( AI \) is radius (length \( r \)), then ratio \( 2r:r = 2:1 \), not \( 1:2 \).

Option D: \( AI:EF \): If \( AI \) is radius (\( r \)) and \( EF \) is diameter (\( 2r \)), then ratio \( r:2r = 1:2 \). Wait, that makes sense. Wait, but wait, is \( EF \) a diameter? Let's see, \( A \) is center, so if \( EF \) passes through \( A \), then it's a diameter. Looking at the diagram, \( EF \) (wait, the points \( E \) and \( F \): \( E \) is on the circle, \( F \) is on the circle, and line \( EF \) passes through \( A \)? Wait, the line from \( E \) through \( A \) to \( F \)? So \( EF \) is a diameter (length \( 2r \)), and \( AI \) is a radius (length \( r \)). So \( AI:EF = r:2r = 1:2 \).

Option E: \( AI/AD \): Both radii, ratio \( 1:1 \).

So the correct ratio is \( AI:EF = 1:2 \), so option D? Wait, no, wait the option is \( AI:EF \), so ratio \( 1:2 \).

Wait, let's confirm: Radius \( r \), diameter \( 2r \). So if \( AI \) is radius (\( r \)) and \( EF \) is diameter (\( 2r \)), then \( AI:EF = r:2r = 1:2 \). So that's the ratio \( 1:2 \).

Step2: Analyze each option

• Option A: \( AF \) and \( AE \) are radii (same length), ratio \( 1:1 \). Eliminate.
• Option B: \( BC \) and \( EF \): Chords, no guarantee ratio \( 1:2 \). Eliminate.
• Option C: \( EF/AI \): If \( EF \) is diameter (\( 2r \)) and \( AI \) is radius (\( r \)), ratio \( 2:1 \). Not \( 1:2 \). Eliminate.
• Option D: \( AI:EF \): \( AI \) (radius \( r \)) and \( EF \) (diameter \( 2r \)), ratio \( r:2r = 1:2 \). Correct.
• Option E: \( AI \) and \( AD \) are radii, ratio \( 1:1 \). Eliminate.

Answer:

D. \( AI:EF \)