a square garden has the end points of A (-7, 6), B (-2, -6), C (10, -1), and D (5, 11). Use the Pythagorean Theorem to find the side length, s, of Square ABCD in feet.
To find the side length, we need to find the distance between any two adjacent points. Let's find the distance between points A and B.
Using the Pythagorean Theorem, we have:
d^2 = (x2 - x1)^2 + (y2 - y1)^2
Where d is the distance between the points (x1, y1) and (x2, y2).
In this case, the points are A(-7, 6) and B(-2, -6).
So, x1 = -7, y1 = 6, x2 = -2, and y2 = -6.
Plugging these values into the formula, we have:
d^2 = (-2 - (-7))^2 + (-6 - 6)^2
= (-2 + 7)^2 + (-6 - 6)^2
= (5)^2 + (-12)^2
= 25 + 144
= 169
Taking the square root of both sides, we get:
d = √169
= 13
Therefore, the distance between points A and B is 13 feet.
Since we know that square ABCD is a square, all sides would have the same length. So, the side length, s, of square ABCD is also 13 feet.
what is the area of the square?
The area of a square can be found by squaring the length of one of its sides.
In this case, the side length of square ABCD is 13 feet.
So, the area of square ABCD is:
Area = side length^2
= 13^2
= 169 square feet.
To find the side length of the square ABCD, we need to calculate the distance between two adjacent vertices of the square using the Pythagorean theorem.
The Pythagorean theorem states that for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In our case, each side of the square is the hypotenuse of a right triangle formed by the coordinates of the adjacent vertices.
Let's calculate the length of side AB first:
1. Calculate the difference of x-coordinates: (x₂ - x₁) = (-2) - (-7) = 5
2. Calculate the difference of y-coordinates: (y₂ - y₁) = (-6) - 6 = -12
3. Apply the Pythagorean theorem:
AB² = (5)² + (-12)²
AB² = 25 + 144
AB² = 169
4. Take the square root of both sides to solve for AB:
AB = √169
AB = 13 feet
Now, let's calculate the length of side BC:
1. Calculate the difference of x-coordinates: (x₂ - x₁) = (10) - (-2) = 12
2. Calculate the difference of y-coordinates: (y₂ - y₁) = (-1) - (-6) = 5
3. Apply the Pythagorean theorem:
BC² = (12)² + (5)²
BC² = 144 + 25
BC² = 169
4. Take the square root of both sides to solve for BC:
BC = √169
BC = 13 feet
Since AB = BC, we can conclude that the length of one side of square ABCD is 13 feet.