9514 1404 393
Answer:
E) 25.5 units²
Step-by-step explanation:
The figure is not carefully drawn, so it is difficult to tell that it is a trapezoid. The "height" of it is BC, the diagonal of a rectangle that is 1 unit by 4 units. The Pythagorean theorem tells you that length is ...
BC = √(1² +4²) = √17
We notice that the "bases" of the trapezoid (CD and AB) are 1 and 2 times this length. The trapezoid area formula tells us the area is ...
A = 1/2(b1 +b2)h
A = 1/2((√17)(1 +2))(√17) = 3/2(17) = 25.5 . . . . square units
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<em>Alternate solution</em>
<em>Pick's Theorem</em> is a theorem that can help you find the area of a polygon when its vertices are all on grid points. It tells you the area is ...
A = i + b/2 -1
where i is the number of grid points interior to the figure, and b is the number of grid points on the boundary. Here, the boundary grid points are those that are labeled plus the one at (-2, 1). Counting the interior grid points gives a total of 24, so the area by Pick's theorem is ...
A = 24 + 5/2 -1 = 25.5 . . . square units
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<em>Additional comment</em>
Once you're familiar with Pick's theorem (and/or any of a few other methods of computing area from coordinates), you see that <em>the area of any polygon whose vertices are grid points will be an integer multiple of 1/2</em>. The only such answer choice here is choice E.
in the given closed interval, so the area is exactly given by the definite integral
