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Determine the coordinates of rectangle ABCD. use the coordinates to compute the area of the rectangle

A(-3,-2) B(4,-2) C(4,1) D(-3,1)

____ units^2

5 answers

To determine the area of rectangle ABCD using its coordinates, we first identify the vertices:

  • A(-3, -2)
  • B(4, -2)
  • C(4, 1)
  • D(-3, 1)

Next, we need to find the lengths of the sides of the rectangle.

  1. Length of AB (or CD): Since points A and B have the same y-coordinate (-2), the distance between them is calculated as follows: \[ AB = x_B - x_A = 4 - (-3) = 4 + 3 = 7 \text{ units} \]

  2. Length of AD (or BC): Since points A and D have the same x-coordinate (-3), the distance between them is calculated as follows: \[ AD = y_D - y_A = 1 - (-2) = 1 + 2 = 3 \text{ units} \]

Now that we have the lengths of the sides of the rectangle, we can compute the area \( A \) using the formula: \[ \text{Area} = \text{Length} \times \text{Width} \] Substituting the lengths we found: \[ \text{Area} = AB \times AD = 7 \times 3 = 21 \text{ units}^2 \]

Thus, the area of rectangle ABCD is \( \text{21 units}^2 \).

use the coordinates to compute the approximate area of the rectangle. Round your answer to the nearest whole number.
A(4,2) B(10,4) C(6,16) D(0,14)

______ units^2

To compute the area of the shape formed by points A(4,2), B(10,4), C(6,16), and D(0,14), we first need to verify whether these points form a rectangle. However, from the coordinates provided, it appears that they are not the vertices of a rectangle.

Instead, we can use the shoelace formula (also known as Gauss's area formula) to calculate the area of a polygon given its vertices in order.

Coordinates of vertices in order:

  • A(4, 2)
  • B(10, 4)
  • C(6, 16)
  • D(0, 14)

The shoelace formula is:

\[ \text{Area} = \frac{1}{2} \left| \sum (x_iy_{i+1} - y_ix_{i+1}) \right| \]

Setting up the coordinates: \[ \begin{array}{c|c} x & y \ \hline 4 & 2 \ 10 & 4 \ 6 & 16 \ 0 & 14 \ 4 & 2 \ \end{array} \]

Calculating the sum:

Step 1: Calculate \( x_iy_{i+1} \)

  • From A to B: \( 4 \cdot 4 = 16 \)
  • From B to C: \( 10 \cdot 16 = 160 \)
  • From C to D: \( 6 \cdot 14 = 84 \)
  • From D to A: \( 0 \cdot 2 = 0 \)

Sum of \( x_iy_{i+1} = 16 + 160 + 84 + 0 = 260 \)

Step 2: Calculate \( y_ix_{i+1} \)

  • From A to B: \( 2 \cdot 10 = 20 \)
  • From B to C: \( 4 \cdot 6 = 24 \)
  • From C to D: \( 16 \cdot 0 = 0 \)
  • From D to A: \( 14 \cdot 4 = 56 \)

Sum of \( y_ix_{i+1} = 20 + 24 + 0 + 56 = 100 \)

Step 3: Calculate the area

Using the shoelace formula: \[ \text{Area} = \frac{1}{2} \left| 260 - 100 \right| = \frac{1}{2} \left| 160 \right| = \frac{160}{2} = 80 \text{ units}^2 \]

Rounding this to the nearest whole number, the area of the polygon formed by points A, B, C, and D is:

80 units².

Fiona draws a triangle on a coordinate grid with vertices at (6,2), (4,6), and (1,2). What is the area in square units of Fiona's triangle?

To find the area of a triangle given its vertices, we can use the formula:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Given the vertices of the triangle:

  • \( A(6, 2) \)
  • \( B(4, 6) \)
  • \( C(1, 2) \)

Here, we assign:

  • \( (x_1, y_1) = (6, 2) \)
  • \( (x_2, y_2) = (4, 6) \)
  • \( (x_3, y_3) = (1, 2) \)

Now we can substitute the values into the formula:

\[ \text{Area} = \frac{1}{2} \left| 6(6 - 2) + 4(2 - 2) + 1(2 - 6) \right| \]

Calculating step-by-step:

  1. Calculate \( 6(6 - 2) = 6(4) = 24 \)
  2. Calculate \( 4(2 - 2) = 4(0) = 0 \)
  3. Calculate \( 1(2 - 6) = 1(-4) = -4 \)

Now plug these results back into the formula:

\[ \text{Area} = \frac{1}{2} \left| 24 + 0 - 4 \right| = \frac{1}{2} \left| 20 \right| = \frac{20}{2} = 10 \]

Therefore, the area of Fiona's triangle is

\[ \boxed{10} \text{ square units}. \]