Question

edison high school is designing a new triangular pennant. the school mascot will be inside a circle, and the possible. answer parts a and b below.
circle radius \t r = 2 cm
circle area \t a_c ≈ \square cm²
type of triangle \t \hfill \bigtriangledown
triangle area \t a_t ≈ \square cm²
ratio of circle area to triangle area \t \square

Explanation:

Step1: Calculate Circle Area

The formula for the area of a circle is \( A_c = \pi r^2 \). Given \( r = 2 \, \text{cm} \), substitute \( r \) into the formula:
\( A_c = \pi (2)^2 = 4\pi \approx 4 \times 3.14 = 12.56 \, \text{cm}^2 \).

Step2: Identify Triangle Type

The triangle has sides 6 cm, 8 cm, 10 cm. Check if \( 6^2 + 8^2 = 10^2 \):
\( 36 + 64 = 100 \) and \( 10^2 = 100 \), so it is a right triangle.

Step3: Calculate Triangle Area

For a right triangle, area is \( A_t = \frac{1}{2} \times \text{base} \times \text{height} \). Using base = 8 cm, height = 6 cm:
\( A_t = \frac{1}{2} \times 8 \times 6 = 24 \, \text{cm}^2 \).

Step4: Calculate Ratio

Ratio of circle area to triangle area is \( \frac{A_c}{A_t} = \frac{12.56}{24} \approx 0.523 \) (or \( \frac{4\pi}{24} = \frac{\pi}{6} \approx 0.523 \)).

Answer:

• Circle Area: \( \boldsymbol{12.56} \, \text{cm}^2 \)
• Type of Triangle: \( \boldsymbol{\text{Right Triangle}} \)
• Triangle Area: \( \boldsymbol{24} \, \text{cm}^2 \)
• Ratio: \( \boldsymbol{\approx 0.523} \) (or \( \boldsymbol{\frac{\pi}{6}} \))