这个题目如何解答---**Question Stem:** 如图，要测定被池塘隔开的A, B两点的距离。可以在AB外选一点C连接AC, BC，并分别找出它们的中点D, E，连接DE。现测得DE = 24m，则AB = ___. **Translated Question Stem:** As shown in the figure, to determine the distance between points A and B, which are separated by a pond. One can choose a point C outside of AB, connect AC and BC, and respectively find their midpoints D and E, then connect DE. Now, it is measured that DE = 24m, then AB = ___. **Chart/Diagram Description:** * **Type:** Geometric figure/Diagram illustrating a practical application of geometry. * **Main Elements:** * **Points:** There are five labeled points: A, B, C, D, E. * **Lines:** * A solid line segment connects C to A. * A solid line segment connects C to B. * A solid line segment connects D to E. * A dashed line segment connects A to B. This dashed line segment passes through an irregular, wavy shape. * **Shapes:** * A triangle is formed by points A, B, and C (implied, with AB being the side across the pond). * A smaller triangle is formed by points C, D, and E. * An irregular, wavy shape is positioned between points A and B, representing a "pond" that separates A and B. * **Labels and Annotations:** * Point D is located on the line segment AC. * Point E is located on the line segment BC. * The text explicitly states that D is the midpoint of AC and E is the midpoint of BC. * The length of segment DE is given as 24m. * **Relative Position:** Point C is positioned such that it forms a triangle ABC where AB is the side that needs to be measured but is obstructed by the pond. Points D and E are inside this triangle, connecting the midpoints of the sides AC and BC, respectively. The segment DE is parallel to AB (implied by the midpoint theorem). * **Direction:** Not applicable.

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We need to measure the distance between points A and B, which are separated by a pond. Since we cannot directly measure across the pond, we use a geometric approach. We choose point C outside the pond, connect it to A and B, find midpoints D and E on these lines, and measure DE which equals 24 meters. 这是一个经典的几何测量问题。当我们需要测量被池塘隔开的两点A和B之间的距离时，可以采用三角形中点连线的方法。具体做法是：在AB外选择一点C，连接AC和BC，分别找出它们的中点D和E，然后连接DE。现在已知DE等于24米，我们需要求出AB的长度。 解决这个问题的关键是三角形中点连线定理。该定理指出：连接三角形两边中点的线段平行于第三边，且长度等于第三边长度的一半。在我们的三角形ABC中，由于D和E分别是AC和BC的中点，所以线段DE平行于AB，且DE等于AB的一半。 现在我们来应用这个定理求解。根据三角形中点连线定理，DE等于AB的一半。已知DE等于24米，我们可以列出等式：24等于二分之一乘以AB。两边同时乘以2，得到AB等于2乘以24，即48米。因此，池塘两端A和B之间的距离是48米。 让我们总结一下这个问题的解决过程。首先，我们识别出这是一个测量被障碍物隔开的两点距离的问题。然后，我们应用了三角形中点连线定理，建立了DE等于AB的一半这个关系。最后，通过简单的代数运算，得出AB等于2倍的DE，即48米。这种方法在实际测量中非常实用，避免了直接跨越障碍物的困难。 Now let's prove why the midpoint theorem works. We start with triangle ABC where D and E are midpoints. To prove that DE equals half of AB, we rotate triangle CDE 180 degrees about point E to create point D prime. This rotation forms parallelogram ABED prime. In a parallelogram, opposite sides are equal, so AB equals ED prime. Since ED prime equals 2 times DE, we conclude that AB equals 2 times DE. Now let's apply our proven relationship to solve the problem. Using the midpoint theorem, we know that DE equals half of AB. Given that DE equals 24 meters, we can write the equation: 24 equals one half times AB. Solving for AB, we multiply both sides by 2, giving us AB equals 48 meters. This elegant geometric solution shows how we can measure an inaccessible distance of 48 meters across the pond by measuring only 24 meters on accessible land. In conclusion, this problem beautifully demonstrates how geometric principles can solve practical challenges. The midpoint theorem transforms what seems like an impossible measurement across a pond into a simple calculation on accessible land. Our answer is 48 meters for the distance AB. This method has wide applications in surveying across rivers and valleys, architecture and construction planning, and navigation and mapping. It's a perfect example of how mathematical theory provides elegant solutions to real-world problems, showing the true power and beauty of geometry in practical problem-solving.