Question

find the side length S of square ABCD using the Pythagorean theorem
A (-7, 6)
b. (-2, -6)
C (10,-1)
D (5, 11)

Answer 1

To find the side length \( S \) of square \( ABCD \), we can use the Pythagorean theorem by calculating the distance between two adjacent vertices.

Let's choose vertices A and B for this calculation. You can also use any other adjacent vertices such as B and C, C and D, or D and A. However, for this example, we will find the distance \( AB \).

The coordinates of points A and B are:
- \( A (-7, 6) \)
- \( B (-2, -6) \)

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Plugging in the coordinates of A and B:

\[
d = \sqrt{((-2) - (-7))^2 + ((-6) - 6)^2}
\]

Calculating the differences:

- \( (-2) + 7 = 5 \)
- \( (-6) - 6 = -12 \)

Now substitute these values into the distance formula:

\[
d = \sqrt{(5)^2 + (-12)^2}
\]
\[
d = \sqrt{25 + 144}
\]
\[
d = \sqrt{169}
\]
\[
d = 13
\]

Thus, the side length \( S \) of square \( ABCD \) is \( 13 \).