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. answer parts a and b below

(image of a triangular pennant with wildcats! and a mascot inside a circle)

(chart and diagram: triangle with two sides 10 cm, height h ≈ 8.7 cm, incircle radius r ≈ 2.9 cm)

circle area ( a_c approx square , \text{cm}^2 )
type of triangle: acute isosceles
triangle area ( a_t approx square , \text{cm}^2 )
ratio of circle area to triangle area: ( square )

Explanation:

Step1: Calculate Circle Area

The formula for the area of a circle is \( A_{c} = \pi r^{2} \). Given \( r \approx 2.9 \) cm, substitute into the formula:
\( A_{c} \approx \pi \times (2.9)^{2} \approx 3.1416 \times 8.41 \approx 26.42 \) (using \( \pi \approx 3.1416 \))

Step2: Calculate Triangle Area

The triangle is equilateral (all sides 10 cm) or acute isosceles. The formula for the area of a triangle is \( A_{t} = \frac{1}{2} \times base \times height \). Given base = 10 cm, height \( h \approx 8.7 \) cm:
\( A_{t} \approx \frac{1}{2} \times 10 \times 8.7 = 43.5 \) \( cm^{2} \)

Step3: Ratio of Circle Area to Triangle Area

Divide the circle area by the triangle area: \( \frac{26.42}{43.5} \approx 0.607 \) (or keep more precise values)

Answer:

• Circle Area: \( \approx 26.42 \) \( cm^{2} \)
• Triangle Area: \( \approx 43.5 \) \( cm^{2} \)
• Ratio: \( \approx 0.607 \) (or based on more precise calculations)