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Problem 677

Points \(\mathrm{A}(-4,-2), \mathrm{B}(2,-2), \mathrm{C}(4,3)\), and \(\mathrm{D}(-2,3)\) are the vertices of quadrilateral \(\mathrm{ABCD}\). (a) Plot these points on graph paper and draw the quadrilateral, (b) What kind of quadrilateral is \(\mathrm{ABCD}\) ? (c) Find the area of quadrilateral \(\mathrm{ABCD}\).

Short Answer

Expert verified
The quadrilateral ABCD is a rectangle, and the area is 42 square units.
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Step 1: Plot the Points

To plot the points, establish a Cartesian plane. Make use of the x and y coordinates given to pinpoint the locations of points A, B, C, and D. The coordinates given are A(-4,-2), B(2,-2), C(4,3), and D(-2,3).

Step 2: Draw the Quadrilateral

Connect points A, B, C, and D to form quadrilateral ABCD. Do note that the points should be connected in the order that they are given.

Step 3: Identify the Type of Quadrilateral

By examining the characteristics of the quadrilateral, we can determine the type. Characteristics can include equal sides, equal angles, parallel sides, etc. However, in the absence of specific measurements, the type of quadrilateral can be determined based on observation from the plot.

Step 4: Calculate the Area of the Quadrilateral

The area of a quadrilateral can be calculated in several ways, but in this case, we'll calculate it by dividing the quadrilateral into two triangles - triangle ABD and triangle BCD. The area of each triangle can be calculated using the formula \((1/2) \times \text{base} \times \text{height}\). The total area of the quadrilateral will be the sum of the areas of these two triangles. To calculate the area, find the lengths of the base and height of each triangle and substitute these values into the formula. For triangle ABD: The coordinates of A, B, and D can be used to calculate this area. The length of the base (BD) can be found using the distance formula: \(BD = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\). The height (h) can be determined using the formula: \(h = y_{2}-y_{1}\). The values obtained can then be substituted into the triangle area formula. Repeat the process above for triangle BCD using the corresponding coordinates of B, C, and D. Sum the areas of both triangles to get the total area of quadrilateral ABCD.

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