Question

edison high school is designing a new triangular pennant. the school mascot will be inside a circle, and the circle must touch each side of the pennant. the circle should fill as much of the pennant as possible. answer parts a and b below.
(image of pennant with wildcats! and a circle with mascot)
(diagram of triangle with two sides 10 cm, one side 8 cm, inradius r = 2.4 cm, height h = 6.2 cm)
table with columns: circle area (a_c = □ cm²), type of triangle (dropdown), triangle area (a_t = □ cm²), ratio of circle area to triangle area (□)
(instructions: type integers or decimals rounded to the nearest thousandth as needed)

Explanation:

Step1: Calculate Circle Area

The formula for the area of a circle is \( A_{c} = \pi r^{2} \). Given \( r = 2.4 \) cm, substitute into the formula:
\( A_{c} = \pi \times (2.4)^{2} \approx 3.1416 \times 5.76 \approx 18.0956 \) \( \text{cm}^2 \) (rounded as needed).

Step2: Identify Triangle Type

The triangle has two sides of length 10 cm and one side of 8 cm, so it is an isosceles triangle (two sides equal).

Step3: Calculate Triangle Area

The formula for the area of a triangle is \( A_{t} = \frac{1}{2} \times \text{base} \times \text{height} \). Here, base = 8 cm, height = 9.2 cm (from the diagram):
\( A_{t} = \frac{1}{2} \times 8 \times 9.2 = 4 \times 9.2 = 36.8 \) \( \text{cm}^2 \).

Step4: Calculate Ratio of Areas

The ratio is \( \frac{A_{c}}{A_{t}} \approx \frac{18.0956}{36.8} \approx 0.4917 \) (rounded to the nearest thousandth as needed).

Answer:

• Circle Area: \( \approx 18.10 \) (or \( 18.096 \)) \( \text{cm}^2 \)
• Type of Triangle: Isosceles Triangle
• Triangle Area: \( 36.8 \) \( \text{cm}^2 \)
• Ratio of Circle Area to Triangle Area: \( \approx 0.492 \) (or \( 0.4917 \))