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Show that the points (1, 1), (4, 5), (0, 8), and (-3, 4) are the vertices of a square, and find its area.
Solution:
To illustrate the problem, it is better to draw the figure as follows:
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| Photo by Math Principles in Everyday Life |
The first thing that we have to do is to get the slope of each sides of the parallelogram using the two point formula as follows
For the slope of AB:
For the slope of BC:
For the slope of CD:
For the slope of AD:
Since the slopes are negative reciprocals to each other
then a parallelogram is either a rectangle or a square. All sides of a rectangle or a square are perpendicular to each other.
Next, we need to get the length of the sides of a rectangle or a square using the distance formula of two points as follows
For the length of AB:
For the length of BC:
For the length of CD:
For the length of AD:
Since the sides of a parallelogram are perpendicular to each other and congruent to each other as well,
then it is a square. A square is a closed figure or quadrilateral whose sides are perpendicular and equal to each other.
Since we know the length of the sides of a square, we can calculate the area as follows
Therefore, the area of a square is
