Question

find the perimeter or circumference and area of each figure. round answers to the nearest tenth, if necessary. 4) a(0, 9) b(4, 9) c(0, 1) area ____ perimeter __ 5) a(2, 3) b(4, 1) area __ perimeter ____

Explanation:

Problem 4 (Triangle)

Step 1: Identify base and height

Points \( A(0,9) \), \( B(4,9) \), \( C(0,1) \).
Base \( AB \): distance between \( A \) and \( B \). Since \( y \)-coordinates are equal, \( AB = 4 - 0 = 4 \).
Height \( AC \): distance between \( A \) and \( C \). Since \( x \)-coordinates are equal, \( AC = 9 - 1 = 8 \).

Step 2: Calculate area of triangle

Formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
\( \text{Area} = \frac{1}{2} \times 4 \times 8 = 16 \).

Step 3: Calculate lengths of sides

• \( AB = 4 \) (already found).
• \( AC = 8 \) (already found).
• \( BC \): use distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) for \( B(4,9) \) and \( C(0,1) \).

\( BC = \sqrt{(4 - 0)^2 + (9 - 1)^2} = \sqrt{16 + 64} = \sqrt{80} \approx 8.94 \).

Step 4: Calculate perimeter

Perimeter = \( AB + AC + BC = 4 + 8 + \sqrt{80} \approx 4 + 8 + 8.94 = 20.94 \approx 20.9 \) (rounded to nearest tenth).

Problem 5 (Circle)

Step 1: Find radius

Center \( A(2,3) \), point \( B(4,1) \). Radius \( r \) is distance between \( A \) and \( B \).
Using distance formula: \( r = \sqrt{(4 - 2)^2 + (1 - 3)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.828 \).

Step 2: Calculate area of circle

Formula: \( \text{Area} = \pi r^2 \)
\( \text{Area} = \pi (2\sqrt{2})^2 = \pi \times 8 \approx 25.1 \) (rounded to nearest tenth).

Step 3: Calculate circumference (perimeter of circle)

Formula: \( C = 2\pi r \)
\( C = 2\pi (2\sqrt{2}) \approx 2 \times 3.1416 \times 2.828 \approx 17.8 \) (rounded to nearest tenth).

Final Answers

Problem 4:

• Area: \( \boldsymbol{16} \)
• Perimeter: \( \boldsymbol{\approx 20.9} \)

Problem 5:

• Area: \( \boldsymbol{\approx 25.1} \)
• Perimeter (Circumference): \( \boldsymbol{\approx 17.8} \)

Answer:

Problem 4 (Triangle)

Step 1: Identify base and height

Points \( A(0,9) \), \( B(4,9) \), \( C(0,1) \).
Base \( AB \): distance between \( A \) and \( B \). Since \( y \)-coordinates are equal, \( AB = 4 - 0 = 4 \).
Height \( AC \): distance between \( A \) and \( C \). Since \( x \)-coordinates are equal, \( AC = 9 - 1 = 8 \).

Step 2: Calculate area of triangle

Formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
\( \text{Area} = \frac{1}{2} \times 4 \times 8 = 16 \).

Step 3: Calculate lengths of sides

• \( AB = 4 \) (already found).
• \( AC = 8 \) (already found).
• \( BC \): use distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) for \( B(4,9) \) and \( C(0,1) \).

\( BC = \sqrt{(4 - 0)^2 + (9 - 1)^2} = \sqrt{16 + 64} = \sqrt{80} \approx 8.94 \).

Step 4: Calculate perimeter

Perimeter = \( AB + AC + BC = 4 + 8 + \sqrt{80} \approx 4 + 8 + 8.94 = 20.94 \approx 20.9 \) (rounded to nearest tenth).

Problem 5 (Circle)

Step 1: Find radius

Center \( A(2,3) \), point \( B(4,1) \). Radius \( r \) is distance between \( A \) and \( B \).
Using distance formula: \( r = \sqrt{(4 - 2)^2 + (1 - 3)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.828 \).

Step 2: Calculate area of circle

Formula: \( \text{Area} = \pi r^2 \)
\( \text{Area} = \pi (2\sqrt{2})^2 = \pi \times 8 \approx 25.1 \) (rounded to nearest tenth).

Step 3: Calculate circumference (perimeter of circle)

Formula: \( C = 2\pi r \)
\( C = 2\pi (2\sqrt{2}) \approx 2 \times 3.1416 \times 2.828 \approx 17.8 \) (rounded to nearest tenth).

Final Answers

Problem 4:

• Area: \( \boldsymbol{16} \)
• Perimeter: \( \boldsymbol{\approx 20.9} \)

Problem 5:

• Area: \( \boldsymbol{\approx 25.1} \)
• Perimeter (Circumference): \( \boldsymbol{\approx 17.8} \)