Solve this---**Question Stem:**
WHAT IS THE AREA OF THE COLORED SQUARE ?
**Chart Description:**
* **Type:** Geometric figure.
* **Main Elements:**
* A large outer square, outlined in black.
* An inner quadrilateral, colored light blue.
* Points marked on each side of the outer square. These points are the vertices of the inner blue quadrilateral.
* Lengths are labeled along the sides of the outer square, indicating the segments created by the vertices of the inner quadrilateral. The lengths are 10 and 5 on the top side, 10 and 5 on the right side, 5 and 10 on the bottom side, and 5 and 10 on the left side. The sum of the segments on each side is 15, indicating the outer square has a side length of 15.
* The vertices of the inner blue figure connect points that divide each side of the outer square into segments of lengths 5 and 10. Specifically, starting from a corner and going clockwise, the segments are 10 and 5, then 10 and 5, then 10 and 5, then 10 and 5 (or starting from a different corner, they could be 5 and 10, etc., but the diagram shows consistent division lengths).
* Based on the placement of the vertices and the labeling, the inner figure appears to be a square formed by connecting points on the sides of the larger square such that each vertex of the inner square is connected to two vertices of the outer square, forming right-angled triangles in the corners. Each of these corner triangles has legs of length 5 and 10.
* **Labels:** Lengths 10 and 5 are labeled along the sides of the outer square. The text "WORLD OF ENGINEERING" appears at the bottom right.
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答案文本
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Let's examine this geometric problem. We have a large square with side length 15, and inside it there's a colored square. The colored square is formed by connecting points on the sides of the outer square. Each side is divided into segments of length 10 and 5, creating four corner triangles.
Now let's focus on the corner triangles. Each corner of the outer square contains a right triangle. These triangles have legs of length 5 and 10. The highlighted triangle shows this clearly - one leg has length 10 and the other has length 5. All four corner triangles have the same dimensions.
Now we can apply the Pythagorean theorem to find the side length of the inner square. The hypotenuse of each corner triangle is actually a side of the inner square. Using the theorem: s squared equals 5 squared plus 10 squared, which gives us s squared equals 25 plus 100, equals 125.
Now we can find the area of the colored square. Since we know that s squared equals 125, and the area of a square is simply the side length squared, the area of the colored square is 125 square units. This is our final answer.
Let's summarize our solution. We identified that the colored square is formed by connecting points on the sides of a larger square, creating four corner triangles with legs of length 5 and 10. Using the Pythagorean theorem, we found that s squared equals 125. Since the area of a square equals the side length squared, the area of the colored square is 125 square units. This completes our solution.