Triangle Area 17B656

1. **State the problem:** We need to compare the areas of triangles $\triangle XDC$ and $\triangle YDC$ formed inside two identical rectangles. 2. **Understand the figure and given information:** Both rectangles are identical, meaning they have the same length and width. 3. **Recall the formula for the area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 4. **Identify the base and height for both triangles:** - Both triangles share the same base $DC$ (the bottom side of the rectangle). - The height is the perpendicular distance from the opposite vertex ($X$ or $Y$) to the base $DC$. 5. **Analyze the positions of points $X$ and $Y$:** - $X$ is near the left of the center on the top side of the left rectangle. - $Y$ is at the center on the top side of the right rectangle. 6. **Since both rectangles are identical, their top sides are equal in length and parallel to $DC$.** - The height for both triangles is the same because the top side is parallel and at the same distance from $DC$. 7. **Compare the bases of the triangles:** - The base for both triangles is the segment $DC$, which is the same length in both rectangles. 8. **Check the horizontal segments forming the triangles:** - $\triangle XDC$ is formed by points $X$, $D$, and $C$. - $\triangle YDC$ is formed by points $Y$, $D$, and $C$. 9. **Since $X$ is closer to $D$ than $Y$ is, the horizontal projection of $X$ onto $DC$ is shorter than that of $Y$.** 10. **Calculate the area of each triangle:** - Area of $\triangle XDC = \frac{1}{2} \times DC \times h_X$ - Area of $\triangle YDC = \frac{1}{2} \times DC \times h_Y$ 11. **Because the height $h$ is the same for both triangles (distance from top side to $DC$), the difference in area depends on the horizontal position of $X$ and $Y$.** 12. **However, both triangles share the same base $DC$ and height (rectangle height), so their areas are equal.** **Final answer:** $$\boxed{\text{The areas of } \triangle XDC \text{ and } \triangle YDC \text{ are equal.}}$$