Question

Use the distance between the rose bushes, which are located at A(−5, 4), B(3, 4), and C(3, −5), to explain where a fourth rose bush should be placed so that it forms a rectangle with the other three bushes.

a) A and B are 8 units apart. B and C are 9 units apart. So the fourth bush should be 9 units to the left of point C and 8 units below point A.
b) A and B are 3 units apart. B and C are 9 units apart. So the fourth bush should be 3 units to the right of point B and 9 units below point C.
c) A and B are 8 units apart. B and C are 9 units apart. So the fourth bush should be 8 units to the right of point C and 9 units below point A.
d) A and B are 3 units apart. B and C are 9 units apart. So the fourth bush should be 3 units to the left of point A and 9 units below point B.

Answer 1

To form a rectangle, the fourth bush D should be placed 8 units to the left of point C and 9 units below point A, resulting in the coordinates (−5, −5) which corresponds to option a).

Step-by-step explanation:

The question involves determining the coordinates for a fourth point D to form a rectangle with three given points A, B, and C. Point A is located at (−5, 4), point B at (3, 4), and point C at (3, −5). To form a rectangle, the fourth point D should be directly horizontally aligned with A and vertically aligned with C.

Since A and B are 8 units apart (3 - (-5) = 8) horizontally and have the same y-coordinate, these two points form one side of the rectangle. B and C are 9 units apart vertically (4 - (-5) = 9) and have the same x-coordinate, forming another side of the rectangle. Therefore, the fourth point D must be 8 units horizontally from C (to match the length of AB) and 9 units vertically from A (to match the length of BC).

Considering this, the fourth bush D should be 8 units to the left of point C (3 - 8 = -5) and 9 units below point A (4 - 9 = -5). Thus, the coordinates for point D are (−5, −5), which matches option a).