Question

a spinner for a board game consists of a cardboard circle with two plastic arrows anchored to its center. the arrows are congruent isosceles triangles connected at their bases, as shown. the base of each triangle measures 2 centimeters and the perimeter of each triangle is 10 centimeters. what is the approximate total area of the plastic triangles on the spinner? herons formula: area = \\(\sqrt{s(s - a)(s - b)(s - c)})

Explanation:

Step1: Find the length of the equal - sides

Let the base of the isosceles triangle $a = 2$ cm and the perimeter $P=10$ cm. Since $P=a + 2b$ (where $b$ is the length of the equal - sides), then $10=2 + 2b$. Solving for $b$ gives $2b=10 - 2=8$, so $b = 4$ cm.

Step2: Calculate the semi - perimeter

The semi - perimeter $s=\frac{a + b + b}{2}=\frac{2+4 + 4}{2}=\frac{10}{2}=5$ cm.

Step3: Use Heron's formula to find the area of one triangle

Using Heron's formula $A=\sqrt{s(s - a)(s - b)(s - b)}$, substitute $s = 5$ cm, $a = 2$ cm, and $b = 4$ cm. Then $A=\sqrt{5(5 - 2)(5 - 4)(5 - 4)}=\sqrt{5\times3\times1\times1}=\sqrt{15}\approx3.87$ square centimeters.

Step4: Find the total area of the two congruent triangles

The total area of the two congruent triangles is $2A\approx2\times3.87 = 7.74\approx7.7$ square centimeters.

Answer:

7.7 square centimeters