Question

What is the area of the rectangle shown on the coordinate plane? Enter your answer in the box. Do not round at any steps. units² A rectangle graphed on a coordinate plane. The horizontal x-axis ranges from negative 1 to 13 in increments of 1. The vertical y-axis ranges from negative 10 to 4 in increments of 1. The vertices of the rectangle lie at begin ordered pair 5 comma 2 end ordered pair, begin ordered pair 2 comma negative 1 end ordered pair, begin ordered pair 9 comma negative 8 end ordered pair and begin ordered pair 12 comma negative 5 end ordered pair.

Answer 1

Answer: 42 square units

Check out the attached images below. In figure 1, I plotted the four points given as A, B, C, D; then connected them to form a rectangle.

Figure 2 shows 4 more points E, F, G, H made so that a larger rectangle forms. This newer larger rectangle entirely encloses the first rectangle. The idea is to find the area of the larger rectangle, and subtract off the areas of the four triangular pieces shown in figure 3. These images are attached below.

area of rectangle EFGH = base*height = 10*10 = 100

area of triangle ABE = base*height/2 = 3*3/2 = 4.5

area of triangle ADH = base*height/2 = 7*7/2 = 24.5

area of triangle CGD = base*height/2 = 3*3/2 = 4.5

area of triangle FCB = base*height/2 = 7*7/2 = 24.5

the four triangles have areas that add up to 4.5+24.5+4.5+24.5 = 58

Subtract this from the area of rectangle EFGH to get 100-58 = 42

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alternatively, you can find the distance from A to B getting roughly 4.24264 (use the distance formula). The distance from B to C is roughly 9.89949

Multiply those two values: 4.24264 * 9.89949 = 41.9999722536 which rounds to 42. There's rounding error based on the fact that the previously mentioned decimal values are approximate.