D is any point on side AC of a triangle ABC with AB = AC, then which of the following is true ?

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In the given problem, we have a triangle ABC where AB = AC, indicating that triangle ABC is an isosceles triangle. Point D is any point on side AC. We need to determine the relationship between the lengths of segments CD and BD. ### Explanation for Option 1: CD < BD 1. **Isosceles Triangle Property**: Since AB = AC, triangle ABC is isosceles with AB and AC as the equal sides. This implies that angles opposite these sides, ∠B and ∠C, are also equal. 2. **Position of Point D**: Point D is located on side AC. Depending on the position of D, the segment CD will always be shorter than BD because: - If D is closer to C, CD is small, and BD spans a larger portion of the triangle, making BD longer. - If D is closer to A, CD approaches zero, while BD approaches the length of AB, which is equal to AC, thus making BD longer. 3. **Triangle Inequality Theorem**: In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Applying this to triangle BCD: - BD + CD > BC - Since AB = AC, and D is on AC, BD will always be greater than CD. Therefore, CD is always less than BD, confirming that Option 1 is correct. ### Explanation for Why Other Options are Incorrect: - **Option 2: CD > BD**: This option suggests that CD is greater than BD, which contradicts the properties of an isosceles triangle and the positioning of point D on side AC. As explained, BD will always be longer than CD due to the triangle's symmetry and the triangle inequality theorem. - **Option 3: CD = BD**: This would imply that point D is equidistant from points B and C, which is not possible unless D coincides with the midpoint of AC, which is not specified in the problem. Even if D were the midpoint, BD would still be longer due to the isosceles nature of the triangle. - **Option 4: None of these**: This option is incorrect because we have already established that CD < BD is a valid and true statement based on the given conditions. In conclusion, the correct answer is Option 1: CD < BD, as it aligns with the geometric properties of the isosceles triangle and the positioning of point D on side AC.