A square garden has the end points listed below. Plot and label these points on the grid:
(-7,6)
(-2,-6)
(10,-1)
(5,11)
2. Connect the points to create Square ABCD.
3. Use the Pythagorean Theorem to find the side length, s, of Square ABCD in feet.
S = ___feet
4. Use the formula, A=s^2. To find the area of Square ABCD
A = ___feet^2
If you plot the given points on the coordinate grid, they would be as follows:
- Point A (-7,6)
- Point B (-2,-6)
- Point C (10,-1)
- Point D (5,11)
You should plot these points on a coordinate plane and then draw lines to connect them in order: A to B, B to C, C to D, and D back to A. This will form the square ABCD.
However, if you carried out the plotting, you would actually notice that these points do not form a perfect square upon connecting them, because the sides will not be equal in length and the angles will not be all 90 degrees. Nevertheless, for the sake of this exercise, let's proceed under the assumption that the points do indeed form a square. We would need to use the distances between the points to find the length of a side of the square.
The distance between two points (x1, y1) and (x2, y2) is defined as:
\(d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\)
Let's choose two adjacent points to calculate one side's length (AB, BC, CD, or DA). We'll calculate the distance between A and B.
\(AB = \sqrt{((-2) - (-7))^2 + ((-6) - 6)^2}\)
\(AB = \sqrt{(5)^2 + (-12)^2}\)
\(AB = \sqrt{25 + 144}\)
\(AB = \sqrt{169}\)
\(AB = 13\) feet
Since it's a square, all sides are equal, so the side length \(s\) of the square is 13 feet.
Now, we can use the area formula for a square:
Area \(A = s^2\)
Area \(= 13^2\)
Area \(= 169\) square feet
Therefore, under the assumption that these points form a square, the side length would be 13 feet, and the area of the square would be 169 square feet. However, please remember that in reality, these points do not form a square.
S=?
A=?
Based on the previous explanation, assuming that the points given do form a square (which they do not in reality), the calculations for the side length \( s \) and the area \( A \) of the square would be as follows:
Side length \( s \) = 13 feet
Area \( A \) = \( s^2 \) = \( 13^2 \) = 169 square feet
So, with the assumption:
\( S = 13 \) feet
\( A = 169 \) square feet
To plot the points on a grid, you will need to graph each point using its x and y coordinates.
1. Start by drawing a grid with labeled x and y axes.
2. Locate the point (-7, 6) on the grid. Begin at the origin (0,0) and count 7 units to the left along the x-axis, then 6 units upward along the y-axis. Mark this point as A.
3. Locate the point (-2, -6) on the grid. Begin at the origin (0,0) and count 2 units to the left along the x-axis, then 6 units downward along the y-axis. Mark this point as B.
4. Locate the point (10, -1) on the grid. Begin at the origin (0,0) and count 10 units to the right along the x-axis, then 1 unit downward along the y-axis. Mark this point as C.
5. Locate the point (5, 11) on the grid. Begin at the origin (0,0) and count 5 units to the right along the x-axis, then 11 units upward along the y-axis. Mark this point as D.
6. Connect the points A, B, C, and D to create Square ABCD.
To find the side length, s, of Square ABCD, you can use the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, you can use the points A and C to find the length of the side of the square.
The horizontal distance between A and C is the difference in their x-coordinates: 10 - (-7) = 17.
The vertical distance between A and C is the difference in their y-coordinates: -1 - 6 = -7.
Now, apply the Pythagorean Theorem:
s^2 = (17)^2 + (-7)^2
s^2 = 289 + 49
s^2 = 338
Taking the square root of both sides, we have:
s = √338
So, the side length, s, of Square ABCD is approximately 18.39 feet (rounded to two decimal places).
To find the area of Square ABCD, you can use the formula A = s^2, where A is the area and s is the side length.
A = (18.39)^2
A = 338.3521
The area of Square ABCD is approximately 338.35 square feet (rounded to two decimal places).