Question

1. which statement below must be true about circle q? a. the distance from u to w is the same as the distance from r to t. b. the distance from u to w is the same as the distance from q to j. c. the distance from r to t is half the distance from q to r. d. the distance from r to t is twice the distance from q to j.

Explanation:

Step1: Recall circle - related concepts

In a circle, the radius is the distance from the center of the circle to any point on the circle. All radii of a circle are equal.

Step2: Analyze each option

• Option A: There is no information given to suggest that the chords \(UW\) and \(RT\) are equal. Chords can have different lengths in a circle.
• Option B: If \(Q\) is the center of the circle and \(J\) is a point on the circle, and we assume that \(U\) and \(W\) are also points on the circle, then the distance from \(U\) to \(W\) (a chord - length) and the distance from \(Q\) to \(J\) (a radius) are not necessarily equal. A chord length can vary, while the radius is a fixed - length from the center to the circle.
• Option C: There is no information given to suggest that the chord \(RT\) and the radius \(QR\) have such a relationship.
• Option D: If \(Q\) is the center of the circle, \(QJ\) is a radius, and if \(RT\) is a diameter (assuming \(R\) and \(T\) are endpoints of a diameter), the diameter of a circle is twice the radius. So the distance from \(R\) to \(T\) (diameter) is twice the distance from \(Q\) to \(J\) (radius).

Answer:

D. The distance from \(R\) to \(T\) is twice the distance from \(Q\) to \(J\).