Question

A square garden has the end points listed below. Plot and label these points on the grid:
A (−7, 6)B (−2, −6)C (10, −1)D (5, 11)

Connect the points to create Square ABCD
.
Use the Pythagorean Theorem to find the side length, s
, of Square ABCD
.
Use the area formula, A = s2
, to find the area of Square ABCD
.

Answer 1

To proceed with your request, we need to first check if the points given form a square and then calculate the relevant properties if they do.

The points are:
- A (-7, 6)
- B (-2, -6)
- C (10, -1)
- D (5, 11)

Let's first find the distances between each pair of points to identify the sides and diagonals:

1. **Calculate the distances:**

Using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

- Distance \( AB \):
\[
d_{AB} = \sqrt{(-2 - (-7))^2 + (-6 - 6)^2} = \sqrt{(5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]

- Distance \( BC \):
\[
d_{BC} = \sqrt{(10 - (-2))^2 + (-1 - (-6))^2} = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13
\]

- Distance \( CD \):
\[
d_{CD} = \sqrt{(5 - 10)^2 + (11 - (-1))^2} = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]

- Distance \( DA \):
\[
d_{DA} = \sqrt{(-7 - 5)^2 + (6 - 11)^2} = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13
\]

2. **Check Diagonal lengths:**

- Distance \( AC \):
\[
d_{AC} = \sqrt{(10 - (-7))^2 + (-1 - 6)^2} = \sqrt{(17)^2 + (-7)^2} = \sqrt{289 + 49} = \sqrt{338}
\]

- Distance \( BD \):
\[
d_{BD} = \sqrt{(5 - (-2))^2 + (11 - (-6))^2} = \sqrt{(7)^2 + (17)^2} = \sqrt{49 + 289} = \sqrt{338}
\]

3. **Establishing characteristics of ABCD:**
- All four sides are equal (13), and the diagonals are equal (\(\sqrt{338}\)), which indicates that ABCD is a square.

4. **Finding the area:**
- The side length \( s \) is \( 13 \).
- Area \( A \) can be calculated as:
\[
A = s^2 = 13^2 = 169
\]

5. **Result summary:**
- **Side length, \( s \)**: \( 13 \)
- **Area, \( A \)**: \( 169 \)

To visualize the square, plot the points A, B, C, and D on a coordinate grid. The square would have vertices at those coordinates, with the connections forming the sides and confirming it meets the properties described above.