C is a circle with radius r. If DE is a chord in circle C, is DE < 6? 1) \(r = 5\). 2) The distance from the center of C to DE is smaller than 4.

Explanation: Given the circle, DE is a chord. We need to know if \(DE < 6\). Statement 1: \(r=5\). The maximum length of a chord is the diameter, \(2r = 10\). Since \(DE\) can be any length up to 10 (e.g., 3, 6, 9), we cannot definitively say if \(DE < 6\). Not sufficient. Statement 2: Let \(d\) be the distance from the center to \(DE\). \(d < 4\). The chord length \(L\) relates to \(r\) and \(d\) by the Pythagorean theorem: \((L/2)^2 + d^2 = r^2\). Since \(r\) is unknown, we cannot determine \(L\). Not sufficient. Together: \(r=5\) and \(d < 4\). From the relation, \((DE/2)^2 + d^2 = 5^2 = 25\). \((DE/2)^2 = 25 - d^2\). Since \(d < 4\), \(d^2 < 16\). So, \(25 - d^2 > 25 - 16 = 9\). \((DE/2)^2 > 9\). \(DE/2 > 3\). \(DE > 6\). Since \(DE\) **must** be greater than \(6\), the answer to the question "is DE < 6?" is definitively "No". Therefore, the statements TOGETHER ARE sufficient. The provided answer/explanation is incorrect as it states: "Statement -2: যদি ধরেনি, center হবে Chord এর distance 4 তখন, Chord এর length হয় 6 কিন্তু যেহেতু distance 4 থেকে ছোট, length অবশ্যই 6 থেকে বেশি।" The problem is data sufficiency, not solving. Together is sufficient to answer a definitive NO.