M25-19

Official Solution:

If the graph of the equation \(|\frac{x}{2}| + |\frac{y}{2}| = 5\) encloses a certain region on the coordinate plane, what is the area of that region?

A. 20
B. 50
C. 100
D. 200
E. 400


First, to simplify the given expression, multiply it by 2: \(|x| + |y| = 10\).

Next, find the \(x\) and \(y\) intercepts of the region (the x-intercept is the value(s) of \(x\) when \(y = 0\), and the y-intercept is the value(s) of \(y\) when \(x = 0\)):

When \(y = 0\), we get \(|x| = 10\), which implies \(x = 10\) and \(x = -10\).

When \(x = 0\), we get \(|y| = 10\), which implies \(y = 10\) and \(y = -10\).

Thus, we have 4 points: (10, 0), (-10, 0), (0, 10), and (0, -10).

When you connect these points, you will see the region enclosed by \(|x| + |y| = 10\):



You can see that the enclosed region forms a square. It is a square because the diagonals of the rectangle are equal (20 and 20) and are also perpendicular bisectors of each other (since they lie on the X and Y axes). Therefore, it must be a square.

Since this square has a diagonal equal to 20, the area of the square can be calculated as \(Area_{square} = \frac{diagonal^2}{2} = \frac{20^2}{2} = 200\).

Alternatively, the side of the square can be calculated as \(\text{Side} = \sqrt{200}\), so the area is equal to \(\text{area} = \text{side}^2 = 200\).
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Answer: D