Question

lian is making a banner for the school gym. she cut three pieces of fabric into congruent isosceles triangles, as shown.
trigonometric area formula: area = \\(\frac{1}{2}ab\sin(c)\\)
to the nearest square inch, how much fabric will she use to make all three triangles?
\\(\circ\\) 139 square inches
\\(\circ\\) 172 square inches
\\(\circ\\) 417 square inches
\\(\circ\\) 515 square inches

Explanation:

Step1: Find area of one triangle

Using the formula \( \text{Area} = \frac{1}{2}ab\sin(C) \), where \( a = 17 \), \( b = 17 \), and \( C = 74^\circ \).
\( \text{Area of one triangle} = \frac{1}{2} \times 17 \times 17 \times \sin(74^\circ) \)
Calculate \( \sin(74^\circ) \approx 0.9613 \)
\( \text{Area of one triangle} \approx \frac{1}{2} \times 17 \times 17 \times 0.9613 \)
\( \approx \frac{1}{2} \times 289 \times 0.9613 \)
\( \approx 144.5 \times 0.9613 \approx 138.9 \) (approx 139)

Step2: Find area of three triangles

Multiply area of one triangle by 3.
\( \text{Total area} = 139 \times 3 = 417 \)

Answer:

417 square inches