Question

use the law of detachment to analyze the following statements
statement 1: if a chord extends through the center of a circle, it is the diameter of the circle.
statement 2: chord ab extends through the center of circle o
which statements are true? check all that apply
□ chord ab is the diameter of circle o
□ all chords are diameters
□ statement 1 is a conditional statement
□ statement 2 relates to the hypothesis of statement 1.
□ chord ab is not the diameter of circle o.

Explanation:

• Chord AB is the diameter of circle O: By the law of detachment, since Statement 1 is a conditional statement (if - then form) where the hypothesis is "a chord extends through the center of a circle" and the conclusion is "it is the diameter of the circle", and Statement 2 satisfies the hypothesis (Chord AB extends through the center of circle O), we can conclude Chord AB is the diameter.
• All chords are diameters: A chord is a line segment with both endpoints on the circle. A diameter is a chord that passes through the center. There are chords (e.g., a chord that does not pass through the center) that are not diameters.
• Statement 1 is a conditional statement: A conditional statement has the form "If \(p\) (hypothesis), then \(q\) (conclusion)". Statement 1 is in the form "If a chord extends through the center of a circle (\(p\)), then it is the diameter of the circle (\(q\))".
• Statement 2 relates to the hypothesis of Statement 1: The hypothesis of Statement 1 is about a chord extending through the center of a circle. Statement 2 says "Chord AB extends through the center of circle O", which is exactly the hypothesis of Statement 1.
• Chord AB is not the diameter of circle O: As we have already used the law of detachment (from Statement 1 and Statement 2) to conclude that Chord AB is the diameter, this statement is false.

Answer:

• Chord AB is the diameter of circle O
• Statement 1 is a conditional statement
• Statement 2 relates to the hypothesis of Statement 1