12 multiple-choice questions, progressively harder.
A pentagon has boundary vertices A(−2,1)A(-2,1)A(−2,1), B(2,−1)B(2,-1)B(2,−1), C(6,2)C(6,2)C(6,2), D(4,5)D(4,5)D(4,5), and E(0,4)E(0,4)E(0,4). What is its ordinary area?
Solution
Correct answer: D
The differences in boundary order are 000, 101010, 222222, 161616, and 888. The last difference comes from E→AE\to AE→A.
The ordinary area is 28 square units.
The product differences for P→QP\to QP→Q, Q→RQ\to RQ→R, and R→PR\to PR→P are 888, −11-11−11, and −19-19−19, respectively. What is the first difference for the order P,R,QP,R,QP,R,Q?
Correct answer: A
The first new edge is P→RP\to RP→R. It reverses R→PR\to PR→P.
The signed area of the triangle in the order T(0,−2)T(0,-2)T(0,−2), U(7,1)U(7,1)U(7,1), V(2,t)V(2,t)V(2,t) is 181818. What is ttt?
The differences are 141414, 7t−27t-27t−2, and −4-4−4.
An edge moves 5 units left. Its two endpoints are above both y=−1y=-1y=−1 and y=−4y=-4y=−4. By how much does its signed trapezoid contribution increase when the reference line changes from y=−1y=-1y=−1 to y=−4y=-4y=−4?
Correct answer: B
Each endpoint height increases by 3, so the average height increases by 3. The signed width is positive 5.
Rectangle corners A(−3,−1)A(-3,-1)A(−3,−1), B(5,−1)B(5,-1)B(5,−1), C(5,3)C(5,3)C(5,3), and D(−3,3)D(-3,3)D(−3,3) are joined in the order B,D,A,CB,D,A,CB,D,A,C and closed at BBB. Which pair crosses?
Correct answer: C
The listed order uses both rectangle diagonals, BDBDBD and ACACAC. Their midpoints agree.
The diagonals meet at that point. The other two segments are opposite sides of the rectangle.
The diagram shows a lattice polygon with vertices A(−1,0)A(-1,0)A(−1,0), B(8,0)B(8,0)B(8,0), C(8,2)C(8,2)C(8,2), D(2,2)D(2,2)D(2,2), E(2,6)E(2,6)E(2,6), and F(−1,6)F(-1,6)F(−1,6) in boundary order. Which expression checks the shoelace result by adding the lower rectangle to the upper rectangle?
The lower rectangle has width 9 and height 2. The upper rectangle runs from horizontal coordinate −1-1−1 to 222 and from height 2 to height 6. Its width is 3 and its height is 4.
A boundary contains the edge from P(1,−2)P(1,-2)P(1,−2) to R(9,2)R(9,2)R(9,2). The point Q(5,0)Q(5,0)Q(5,0) is inserted between them in the list. What is the sum of the two new product differences?
The two new edges go from PPP to QQQ and from QQQ to RRR. Compute both differences.
Their sum is 20, the same as the difference for the original edge.
A map uses 3 meters for each coordinate unit on both axes. The signed area from the boundary coordinates is −17/2-17/2−17/2 square coordinate units. What is the actual ordinary area in square meters?
The ordinary area is 17/217/217/2 square coordinate units. Each unit square has side length 3 meters and area 9 square meters.
The area is 153/2153/2153/2 square meters.
In the trapezoid derivation, which two contributions cancel the endpoint product xihix_i h_ixihi at vertex PiP_iPi?
The incoming edge ends at PiP_iPi, so its endpoint difference has −xihi-x_i h_i−xihi. The outgoing edge starts at PiP_iPi, so its endpoint difference has +xihi+x_i h_i+xihi.
A strip of width 2 contains two separate pieces of a polygon. Each piece has a straight upper edge and a straight lower edge across the entire strip. For the lower piece, the vertical gaps at the two ends are 3 and 3. For the upper piece, the gaps are 2 and 5. What area lies in the strip?
With straight edges the gap changes linearly across the strip, so its average is the average of the two end gaps. Multiply each average by the strip width and add the results.
A vertical edge goes upward along x=4x=4x=4. Its starting and ending heights above the reference line are 3 and 8. Its cross-product difference in those height coordinates is 20. What endpoint difference accompanies it in the trapezoid derivation?
The trapezoid contribution is zero because the signed width is zero. Twice that contribution is the sum of the given cross-product difference and the endpoint difference bib_ibi.
Some product differences of a simple polygon in boundary order are negative, but their total is positive. Which claim follows?
The positive total is twice the signed area.
Individual product differences can have either sign. The positive total identifies counterclockwise orientation.
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