Question

find the perimeter or circumference and area of each figure. round answers to the nearest tenth, if necessary.
a) a(0, 3), b(4, 3), c(0, 1) triangle on coordinate plane
b) a(2, 3), b(4, 1) circle on coordinate plane
area ______
perimeter ______
area ______
perimeter ______

Explanation:

Part a) (Triangle)

Step 1: Identify the coordinates and side lengths

• Points: \( A(0, 3) \), \( B(4, 3) \), \( C(0, 1) \)
• \( AB \): Horizontal distance (same \( y \)-coordinate). Length \( = 4 - 0 = 4 \)
• \( AC \): Vertical distance (same \( x \)-coordinate). Length \( = 3 - 1 = 2 \)
• \( BC \): Use distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

\( BC = \sqrt{(4 - 0)^2 + (3 - 1)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \approx 4.47 \)

Step 2: Calculate Area

Area of triangle \( = \frac{1}{2} \times \text{base} \times \text{height} \)
Base \( AB = 4 \), Height \( AC = 2 \)
Area \( = \frac{1}{2} \times 4 \times 2 = 4 \)

Step 3: Calculate Perimeter

Perimeter \( = AB + AC + BC = 4 + 2 + 2\sqrt{5} \approx 4 + 2 + 4.47 = 10.47 \)

Part b) (Circle)

Step 1: Find the radius

• Center \( A(2, 3) \), Point \( B(4, 1) \)
• Radius \( r = \) distance between \( A \) and \( B \)

\( r = \sqrt{(4 - 2)^2 + (1 - 3)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.83 \)

Step 2: Calculate Area

Area of circle \( = \pi r^2 = \pi (2\sqrt{2})^2 = \pi \times 8 = 8\pi \approx 25.13 \)

Step 3: Calculate Circumference (Perimeter)

Circumference \( = 2\pi r = 2\pi (2\sqrt{2}) = 4\sqrt{2}\pi \approx 17.77 \)

Final Answers

a) Triangle:

• Area: \( \boldsymbol{4} \)
• Perimeter: \( \boldsymbol{6 + 2\sqrt{5} \approx 10.47} \)

b) Circle:

• Area: \( \boldsymbol{8\pi \approx 25.13} \)
• Circumference (Perimeter): \( \boldsymbol{4\sqrt{2}\pi \approx 17.77} \)

Answer:

Part a) (Triangle)

Step 1: Identify the coordinates and side lengths

• Points: \( A(0, 3) \), \( B(4, 3) \), \( C(0, 1) \)
• \( AB \): Horizontal distance (same \( y \)-coordinate). Length \( = 4 - 0 = 4 \)
• \( AC \): Vertical distance (same \( x \)-coordinate). Length \( = 3 - 1 = 2 \)
• \( BC \): Use distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

\( BC = \sqrt{(4 - 0)^2 + (3 - 1)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \approx 4.47 \)

Step 2: Calculate Area

Area of triangle \( = \frac{1}{2} \times \text{base} \times \text{height} \)
Base \( AB = 4 \), Height \( AC = 2 \)
Area \( = \frac{1}{2} \times 4 \times 2 = 4 \)

Step 3: Calculate Perimeter

Perimeter \( = AB + AC + BC = 4 + 2 + 2\sqrt{5} \approx 4 + 2 + 4.47 = 10.47 \)

Part b) (Circle)

Step 1: Find the radius

• Center \( A(2, 3) \), Point \( B(4, 1) \)
• Radius \( r = \) distance between \( A \) and \( B \)

\( r = \sqrt{(4 - 2)^2 + (1 - 3)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.83 \)

Step 2: Calculate Area

Area of circle \( = \pi r^2 = \pi (2\sqrt{2})^2 = \pi \times 8 = 8\pi \approx 25.13 \)

Step 3: Calculate Circumference (Perimeter)

Circumference \( = 2\pi r = 2\pi (2\sqrt{2}) = 4\sqrt{2}\pi \approx 17.77 \)

Final Answers

a) Triangle:

• Area: \( \boldsymbol{4} \)
• Perimeter: \( \boldsymbol{6 + 2\sqrt{5} \approx 10.47} \)

b) Circle:

• Area: \( \boldsymbol{8\pi \approx 25.13} \)
• Circumference (Perimeter): \( \boldsymbol{4\sqrt{2}\pi \approx 17.77} \)