Question

1. find the area and perimeter of a rectangle with the following coordinates.

p(-6, 4), q(7, 4), r(7, -6), and s(-6, -6)

brainworks

1. the diagram shows a rectangle abcd. ad cuts the x-axis at m and bc cuts the x-axis at n. if the area of the rectangle is 60 cm², and 2/3 of the area is above the x-axis, find the coordinates of a, b, c, and d.

diagram: rectangle abcd with ad and bc cutting x-axis at m(-3,0) and n(7,0) respectively, y-axis in the middle

Explanation:

Problem 11: Find the area and perimeter of a rectangle with coordinates \( P(-6, 4) \), \( Q(7, 4) \), \( R(7, -6) \), and \( S(-6, -6) \)

Step 1: Find the length of the rectangle

The length is the distance between \( P(-6, 4) \) and \( Q(7, 4) \). Since the y-coordinates are the same, we calculate the difference in x-coordinates:
\( \text{Length} = |7 - (-6)| = |7 + 6| = 13 \)

Step 2: Find the width of the rectangle

The width is the distance between \( Q(7, 4) \) and \( R(7, -6) \). Since the x-coordinates are the same, we calculate the difference in y-coordinates:
\( \text{Width} = |-6 - 4| = |-10| = 10 \)

Step 3: Calculate the area of the rectangle

The formula for the area of a rectangle is \( \text{Area} = \text{length} \times \text{width} \):
\( \text{Area} = 13 \times 10 = 130 \) square units

Step 4: Calculate the perimeter of the rectangle

The formula for the perimeter of a rectangle is \( \text{Perimeter} = 2 \times (\text{length} + \text{width}) \):
\( \text{Perimeter} = 2 \times (13 + 10) = 2 \times 23 = 46 \) units

Step 1: Find the length of the rectangle (distance between \( M(-3, 0) \) and \( N(7, 0) \))

The length is the horizontal distance between \( M \) and \( N \):
\( \text{Length} = |7 - (-3)| = |7 + 3| = 10 \) cm

Step 2: Find the total height of the rectangle

The area of the rectangle is \( 60 \, \text{cm}^2 \), and \( \text{Area} = \text{length} \times \text{total height} \). Solving for total height:
\( 60 = 10 \times \text{total height} \implies \text{total height} = \frac{60}{10} = 6 \) cm

Step 3: Find the height above and below the x-axis

\( \frac{2}{3} \) of the area is above the x-axis, so the height above the x-axis is \( \frac{2}{3} \times 6 = 4 \) cm, and the height below is \( 6 - 4 = 2 \) cm (or \( \frac{1}{3} \times 6 = 2 \) cm).

Step 4: Determine coordinates

• \( M(-3, 0) \) and \( N(7, 0) \) are on the x-axis.
• \( A \) and \( D \) lie above and below \( M \); \( B \) and \( C \) lie above and below \( N \).

• \( A \): Same x as \( M(-3) \), y = height above x-axis (\( 4 \)): \( A(-3, 4) \)
• \( B \): Same x as \( N(7) \), y = height above x-axis (\( 4 \)): \( B(7, 4) \)
• \( C \): Same x as \( N(7) \), y = -height below x-axis (\( -2 \)): \( C(7, -2) \)
• \( D \): Same x as \( M(-3) \), y = -height below x-axis (\( -2 \)): \( D(-3, -2) \)

Answer:

• Area: \( \boldsymbol{130} \) square units
• Perimeter: \( \boldsymbol{46} \) units

Problem 12: Find the coordinates of \( A, B, C, D \) for rectangle \( ABCD \)