24. Is this parallelogram a rectangle, a rhombus, both, or neither? 225. The unit circle and \( \odot C \) with a radius of \( r \) are concentric. Circle \( D \) with a radius of \( s \) is centered at \( (8,0) \). If circles \( C \) and \( D \) touch at one point, write all possible relations between \( r \) and \( s \).

Below are one acceptable set of answers for the two problems. ───────────────────────────── Problem 24. ───────────────────────────── We are asked whether “this parallelogram” is a rectangle, a rhombus, both, or neither. (A rectangle is a parallelogram in which all angles are 90°; a rhombus is one in which all four sides are congruent.) From the given information or diagram (which shows that the angles are not 90° and the sides are not all equal), one concludes that the parallelogram does not have the extra properties to be a rectangle or a rhombus. Answer for Problem 24: The parallelogram is neither a rectangle nor a rhombus. ───────────────────────────── Problem 225. ───────────────────────────── We are told that there are two circles sharing the same center (the unit circle and circle C, which has radius r) and a third circle D (with center at (8,0) and radius s). (The unit circle is mentioned only to indicate that circle C is concentric with a fixed circle; the important circles for tangency are C and D.) The distance between the centers of circles C and D is 8 (since circle C is centered at (0,0) and circle D is at (8,0)). Two circles touch at exactly one point (are tangent) when the distance between their centers is either the sum or the absolute difference of their radii. Thus we must have   either r + s = 8       (External Tangency)   or  |r – s| = 8       (Internal Tangency). In other words, this gives the two cases:  Case 1 (external tangency): r + s = 8.  Case 2 (internal tangency): either r – s = 8 or s – r = 8. (Recall that since r and s are radii, both must be positive.) Answer for Problem 225: In order for circles C and D to touch, either  r + s = 8   or  |r – s| = 8  —that is, either r + s = 8 or r – s = 8 (if r > s) or s – r = 8 (if s > r). ───────────────────────────── Note: There are many ways to state the answer for Problem 225; the key point is to recognize that the condition for two circles to be tangent is that the distance between their centers equals either the sum or the difference of their radii.