Use the Pythagorean theorem to find the side length s of square A (-7,6) B (-2,-6) C (10,-1) D (5,11)
To find the side length of square A, we can use the distance formula to find the lengths of the sides of the square and then apply the Pythagorean theorem.
First, let's find the lengths of the sides of the square using the distance formula.
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Between points A and B:
d_AB = √((-2 - (-7))^2 + (-6 - 6)^2)
= √(5^2 + 12^2)
= √(25 + 144)
= √169
= 13
Between points B and C:
d_BC = √((10 - (-2))^2 + (-1 - (-6))^2)
= √(12^2 + 5^2)
= √(144 + 25)
= √169
= 13
Between points C and D:
d_CD = √((5 - 10)^2 + (11 - (-1))^2)
= √((-5)^2 + 12^2)
= √(25 + 144)
= √169
= 13
Between points D and A:
d_DA = √((-7 - 5)^2 + (6 - 11)^2)
= √((-12)^2 + (-5)^2)
= √(144 + 25)
= √169
= 13
Now that we have the lengths of all sides, we can see that all of them are equal to 13, which means that all sides of square A have a length of 13 units.
To find the side length (s) of square ABCD using the Pythagorean theorem, we need to find the distance between two diagonal corners of the square.
Step 1: Find the distance between point A (-7,6) and point C (10,-1).
The formula for the distance between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of points A and C into the formula:
d = √((10 - (-7))^2 + (-1 - 6)^2)
= √((17)^2 + (-7)^2)
= √(289 + 49)
= √338
= 18.38 (rounded to two decimal places)
Step 2: Divide the distance by √2 to find the side length of the square.
Since the diagonal of a square divides it into two congruent right-angled triangles, the side length of the square can be found by dividing the length of the diagonal by √2.
s = d / √2
= 18.38 / √2
= 18.38 / 1.41 (rounded to two decimal places)
= 13.02 (rounded to two decimal places)
Therefore, the side length (s) of square ABCD is approximately 13.02 units.
To use the Pythagorean theorem to find the side length of square A, we first need to determine the length of one of its sides. Let's use points A and B to find one side length.
1. Find the distance between points A and B using the distance formula:
- The distance formula is given by: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Here, (x1, y1) = (-7, 6) and (x2, y2) = (-2, -6)
- Plugging the values into the formula: d = sqrt((-2 - (-7))^2 + (-6 - 6)^2)
Simplifying, d = sqrt(5^2 + (-12)^2) = sqrt(25 + 144) = sqrt(169) = 13.
2. Since square A has four equal sides, we can conclude that each side length is 13 units.
Therefore, the side length "s" of square A is 13 units.