Question

use the coordinate grid below to answer the question. which four points would be the vertices of a square?

Explanation:

Step1: Identify coordinates of points

First, we find the coordinates of each point:

• \( A(-4, 4) \)
• \( B(2, 4) \)
• \( C(5, 4) \)
• \( D(-4, 1) \)
• \( E(5, 1) \)
• \( F(-4, -2) \)
• \( G(2, -2) \)
• \( H(5, -2) \)

Step2: Check for square properties (equal sides, right angles)

A square has four equal sides and four right angles. Let's check the vertical and horizontal distances.
For a square, the horizontal and vertical distances between consecutive vertices should be equal, and the sides should be perpendicular (horizontal and vertical lines are perpendicular).

Let's consider points \( A(-4, 4) \), \( B(2, 4) \), \( G(2, -2) \), \( F(-4, -2) \):

• Distance \( AB \): \( |2 - (-4)| = 6 \) (horizontal distance, since y-coordinates are same)
• Distance \( BG \): \( |-2 - 4| = 6 \) (vertical distance, since x-coordinates are same)
• Distance \( GF \): \( |-4 - 2| = 6 \) (horizontal distance, since y-coordinates are same)
• Distance \( FA \): \( |4 - (-2)| = 6 \) (vertical distance, since x-coordinates are same)

Also, the angles between \( AB \) (horizontal) and \( BG \) (vertical) is \( 90^\circ \), same for other angles.

Alternatively, check \( B(2, 4) \), \( C(5, 4) \), \( H(5, -2) \), \( G(2, -2) \):

• Distance \( BC \): \( |5 - 2| = 3 \)
• Distance \( CH \): \( |-2 - 4| = 6 \) (not equal, so not square)

Check \( A(-4, 4) \), \( D(-4, 1) \), \( E(5, 1) \), \( C(5, 4) \):

• Distance \( AD \): \( |1 - 4| = 3 \)
• Distance \( DE \): \( |5 - (-4)| = 9 \) (not equal, so not square)

Check \( A(-4, 4) \), \( B(2, 4) \), \( G(2, -2) \), \( F(-4, -2) \):

• All sides are length 6 (horizontal/vertical distances), and angles are \( 90^\circ \), so this is a square.

Answer:

The four points that are the vertices of a square are \( A(-4, 4) \), \( B(2, 4) \), \( G(2, -2) \), and \( F(-4, -2) \) (or other valid square like \( B(2, 4) \), \( C(5, 4) \), \( H(5, -2) \), \( G(2, -2) \) if we consider side length 3? Wait, no, earlier calculation for \( B, C, H, G \):

Wait, recalculating \( B(2,4) \), \( C(5,4) \): distance is \( 5 - 2 = 3 \)

\( C(5,4) \) to \( H(5,-2) \): distance is \( 4 - (-2) = 6 \) (not equal to 3, so not square)

Wait, my first check: \( A(-4,4) \) to \( B(2,4) \): \( 2 - (-4) = 6 \)

\( B(2,4) \) to \( G(2,-2) \): \( 4 - (-2) = 6 \)

\( G(2,-2) \) to \( F(-4,-2) \): \( 2 - (-4) = 6 \)

\( F(-4,-2) \) to \( A(-4,4) \): \( 4 - (-2) = 6 \)

And the sides are horizontal and vertical, so angles are 90 degrees. So that's a square with side length 6.

Alternatively, \( D(-4,1) \), \( E(5,1) \), \( H(5,-2) \), \( F(-4,-2) \):

\( D \) to \( E \): \( 5 - (-4) = 9 \)

\( E \) to \( H \): \( 1 - (-2) = 3 \) (not equal, so no)

Wait, maybe the intended square is \( B(2,4) \), \( C(5,4) \), \( H(5,-2) \), \( G(2,-2) \)? Wait no, side lengths would be 3 (horizontal) and 6 (vertical), which is a rectangle, not square.

Wait, another set: \( A(-4,4) \), \( D(-4,1) \), \( G(2,-2) \), \( B(2,4) \)? No, distances not equal.

Wait, maybe I made a mistake. Let's list all coordinates again:

• \( A \): x=-4, y=4 → (-4,4)
• \( B \): x=2, y=4 → (2,4)
• \( C \): x=5, y=4 → (5,4)
• \( D \): x=-4, y=1 → (-4,1)
• \( E \): x=5, y=1 → (5,1)
• \( F \): x=-4, y=-2 → (-4,-2)
• \( G \): x=2, y=-2 → (2,-2)
• \( H \): x=5, y=-2 → (5,-2)

Now, check \( B(2,4) \), \( G(2,-2) \), \( H(5,-2) \), \( C(5,4) \):

• \( B \) to \( G \): vertical distance: \( 4 - (-2) = 6 \)
• \( G \) to \( H \): horizontal distance: \( 5 - 2 = 3 \)
• Not equal, so rectangle.

Check \( A(-4,4) \), \( B(2,4) \), \( G(2,-2) \), \( F(-4,-2) \):

• \( A \) to \( B \): horizontal, length 6 (2 - (-4) = 6)
• \( B \) to \( G \): vertical, length 6 (4 - (-2) = 6)
• \( G \) to \( F \): horizontal, length 6 (2 - (-4) = 6)
• \( F \) to \( A \): vertical, length 6 (4 - (-2) = 6)
• All sides equal, and angles 90 degrees (since horizontal and vertical lines are perpendicular). So this is a square.

So the four points are \( A \), \( B \), \( G \), \( F \).