AMC8 2013

AMC8 2013 · Q24

AMC8 2013 · Q24. It mainly tests Area & perimeter, Coordinate geometry.

Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ and $HE$, respectively. What is the ratio of the area of the shaded pentagon $AIJCB$ to the sum of the areas of the three squares?

正方形 $ABCD$、$EFGH$ 和 $GHIJ$ 面积相等。点 $C$ 和 $D$ 分别是边 $IH$ 和 $HE$ 的中点。阴影五边形 $AIJCB$ 的面积与三个正方形面积之和的比是多少？

(A) \frac{1}{4} \frac{1}{4}

(B) \frac{7}{24} \frac{7}{24}

(D) \frac{3}{8} \frac{3}{8}

(E) \frac{5}{12} \frac{5}{12}

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let point X be the point that intersects line EI. We can split our original triangle into trapezoid ABCX and triangle XIJ. WLOG (Without Loss Of Generality), AB equals 1 unit. Then since, X is the midpoint of line AJ, and point A is 1.5 units horizontally from J, the midpoint X will be 0.75 units away horizontally from A and thus 0.25 units horizontally from C. Therefore XC equals 0.25 units. Using the area formulas for trapezoids and triangles, we calculate the area of ABCX to be 0.625 and XIJ to be 0.375. The combined areas (which are equivalent to our original triangle) equal 1. Therefore, the ratio of the area of the hexagon to the three squares is 1:3 because the area of the three squares is 3. The answer is $\boxed{\textbf{(C)}\ \frac {1}{3}}$-~TheNerdwhoIsNerdy. Edits for Clarity and Accuracy by RamanujanIsBetter

设点 X 为直线 EI 的交点。我们可以将原三角形分成梯形 ABCX 和三角形 XIJ。假设 AB 长为 1 单位。由于 X 是线段 AJ 的中点，点 A 距 J 水平 1.5 单位，因此中点 X 距 A 水平 0.75 单位，距 C 水平 0.25 单位。因此 XC = 0.25 单位。使用梯形和三角形的面积公式，计算 ABCX 面积为 0.625，XIJ 面积为 0.375。总面积（相当于原三角形）为 1。因此，六边形面积与三个正方形的比为 1:3，因为三个正方形面积总和为 3。答案是 $\boxed{\textbf{(C)}\ \frac {1}{3}}$。

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