AMC 8 Daily Practice Round 8

Complete problem set with solutions and individual problem pages

Problem 21 Easy

In a large square with side length not exceeding 8~\text{cm}, as shown in the figure, three square sheets of paper, each with an area of 20\ \text{cm}^2, are placed inside. The total area covered by these three sheets is 44\ \text{cm}^2. What is the area of the large square in square centimeters?

  • A.

    44

  • B.

    47

  • C.

    49.8

  • D.

    51.2

  • E.

    53.4

Answer:D

The red square is translated so that its left side coincides with the left side of the large square. The total area covered by the three squares remains unchanged.

At this point, the large square is divided into four parts. The yellow square has an area of 20\ \text{cm}^2.

The exposed rectangular parts of the red and blue squares have equal areas, and their total area is 44 - 20 = 24\ \text{cm}^2.

Therefore, each of the red and blue rectangles has an area of 12\ \text{cm}^2.

Let the uncovered area in the upper-right corner be x\ \text{cm}^2.

Then \frac{12}{20} = \frac{x}{12}, \quad x = \frac{12 \times 12}{20} = 7.2.

Thus, the area of the large square is 44 + 7.2 = 51.2 \ \text{cm}^2.

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