Question

A rectangular constellation formed in Galaxy 1 is plotted in the first coordinate plane, and another rectangular constellation formed in Galaxy 2 is plotted in the second coordinate plane. Do you think both have the same areas? Justify your answer with appropriate reasons.

• A. Yes, both have the same area because both have the same lengths.
• B. Yes, both have the same area because the product of the length and width are the same for both rectangles.
• C. No, both have different areas because both rectangles are in different quadrants.
• D. No, both have different areas because both have different side lengths.

Answer 1

The correct option is B Yes, both have the same area because the product of the length and width are the same for both rectangles.

The coordinates of the rectangle ABCD are A(3, 8), B(7, 8), C(7, 3), and D(3, 3).
Length of the rectangle ABCD = Side AB of the figure
Width of the rectangle ABCD = Side BC of the figure

Coordinates		Distance
A(3, 8) and B(7, 8)	y-coordinates are the same for both the points; the absolute value of the difference of x-coordinates will give the length AB	Length = |3 − 7| = |−4| = 4 units
B(7, 8) and C(7, 3)	x-coordinates are the same for both the points; the absolute value of the difference of y-coordinates will give the length BC.	Length = |8 − 3| = |5| = 5 units

Side AB = side CD = 4 units
Side BC = side AD = 5 units
Area of the rectangle $=Length\times Width$
$=4\times 5$
$=20$ sq units
The coordinates of the rectangle PQRS are P(−8, −3), Q(−3, −3), R(−3, −7), and S(−8, −7).
Length of the rectangle PQRS = side PQ of the figure
Width of the rectangle PQRS = side QR of the figure

Coordinates		Distance
P(−8, −3) and Q(−3, −3)	y-coordinates are the same for both the points; the absolute value of the difference of x-coordinates will give the length PQ.	Length = |−8 − (−3)| = |−8 + 3| = |−5| = 5 units
Q(−3, −3) and R(−3, −7)	x-coordinates are the same for both the points; the absolute value of the difference of y-coordinates will give the length QR.	Length = |−3 − (−7)| = |−3 + 7| = |4| = 4 units

Side PQ = SR = 5 units
Side QR = PS = 4 units
Area of the rectangle $=Length\times width$
$=4\times 5$
$=20$ sq units
The area of both the rectangles ABCD and PQRS are the same, as the product of length and width are the same for both the rectangles, but the length AB is not equal to length PQ.

Option D is correct.