Question

3 find the perimeter of the shape in the coordinate plane. round to the nearest hundredth. coordinates of point a: ( , ) coordinates of point b: ( , ) coordinates of point c: ( , ) \\( \overline{ab} = \\) \\( \overline{bc} = \\) \\( \overline{ca} = \\) perimeter:

Explanation:

To solve this, we first determine the coordinates of points \( A \), \( B \), and \( C \) from the grid. Let's assume the grid has each square with side length 1.

Step 1: Identify Coordinates

• Point \( A \): From the origin (0,0), moving left 3 units (x = -3) and up 2 units (y = 2). So \( A(-3, 2) \).
• Point \( B \): Moving right 1 unit (x = 1) and down 1 unit (y = -1). So \( B(1, -1) \).
• Point \( C \): Moving right 2 units (x = 2) and up 1 unit (y = 1). So \( C(2, 1) \).

Step 2: Calculate Lengths Using Distance Formula

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Length of \( \overline{AB} \):

Points \( A(-3, 2) \) and \( B(1, -1) \):
\[
AB = \sqrt{(1 - (-3))^2 + (-1 - 2)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]

Length of \( \overline{BC} \):

Points \( B(1, -1) \) and \( C(2, 1) \):
\[
BC = \sqrt{(2 - 1)^2 + (1 - (-1))^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.24
\]

Length of \( \overline{CA} \):

Points \( C(2, 1) \) and \( A(-3, 2) \):
\[
CA = \sqrt{(-3 - 2)^2 + (2 - 1)^2} = \sqrt{(-5)^2 + (1)^2} = \sqrt{25 + 1} = \sqrt{26} \approx 5.10
\]

Step 3: Calculate Perimeter

Perimeter is the sum of all sides:
\[
\text{Perimeter} = AB + BC + CA = 5 + \sqrt{5} + \sqrt{26} \approx 5 + 2.24 + 5.10 = 12.34
\]

Final Answers

• Coordinates of \( A \): \( \boldsymbol{(-3, 2)} \)
• Coordinates of \( B \): \( \boldsymbol{(1, -1)} \)
• Coordinates of \( C \): \( \boldsymbol{(2, 1)} \)
• \( \overline{AB} \): \( \boldsymbol{5} \)
• \( \overline{BC} \): \( \boldsymbol{\approx 2.24} \)
• \( \overline{CA} \): \( \boldsymbol{\approx 5.10} \)
• Perimeter: \( \boldsymbol{\approx 12.34} \)

Answer:

To solve this, we first determine the coordinates of points \( A \), \( B \), and \( C \) from the grid. Let's assume the grid has each square with side length 1.

Step 1: Identify Coordinates

• Point \( A \): From the origin (0,0), moving left 3 units (x = -3) and up 2 units (y = 2). So \( A(-3, 2) \).
• Point \( B \): Moving right 1 unit (x = 1) and down 1 unit (y = -1). So \( B(1, -1) \).
• Point \( C \): Moving right 2 units (x = 2) and up 1 unit (y = 1). So \( C(2, 1) \).

Step 2: Calculate Lengths Using Distance Formula

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Length of \( \overline{AB} \):

Points \( A(-3, 2) \) and \( B(1, -1) \):
\[
AB = \sqrt{(1 - (-3))^2 + (-1 - 2)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]

Length of \( \overline{BC} \):

Points \( B(1, -1) \) and \( C(2, 1) \):
\[
BC = \sqrt{(2 - 1)^2 + (1 - (-1))^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.24
\]

Length of \( \overline{CA} \):

Points \( C(2, 1) \) and \( A(-3, 2) \):
\[
CA = \sqrt{(-3 - 2)^2 + (2 - 1)^2} = \sqrt{(-5)^2 + (1)^2} = \sqrt{25 + 1} = \sqrt{26} \approx 5.10
\]

Step 3: Calculate Perimeter

Perimeter is the sum of all sides:
\[
\text{Perimeter} = AB + BC + CA = 5 + \sqrt{5} + \sqrt{26} \approx 5 + 2.24 + 5.10 = 12.34
\]

Final Answers

• Coordinates of \( A \): \( \boldsymbol{(-3, 2)} \)
• Coordinates of \( B \): \( \boldsymbol{(1, -1)} \)
• Coordinates of \( C \): \( \boldsymbol{(2, 1)} \)
• \( \overline{AB} \): \( \boldsymbol{5} \)
• \( \overline{BC} \): \( \boldsymbol{\approx 2.24} \)
• \( \overline{CA} \): \( \boldsymbol{\approx 5.10} \)
• Perimeter: \( \boldsymbol{\approx 12.34} \)