Question

write your answer as a whole number or a decimal rounded to the nearest hundredth. square miles what is the area of the shaded region?

Explanation:

Step1: Calculate the area of the larger triangle

Use Heron's formula \(s=\frac{a + b+ c}{2}\), where \(a = 25\), \(b = 34\), \(c = 25\).
\(s=\frac{25+34 + 25}{2}=\frac{84}{2}=42\)
Area \(A_1=\sqrt{s(s - a)(s - b)(s - c)}=\sqrt{42(42 - 25)(42 - 34)(42 - 25)}=\sqrt{42\times17\times8\times17}=\sqrt{97008}=311.46\)

Step2: Calculate the area of the smaller triangle

Use Heron's formula \(s=\frac{a + b + c}{2}\), where \(a = 17\), \(b = 23\), \(c = 23\).
\(s=\frac{17+23+23}{2}=\frac{63}{2}=31.5\)
Area \(A_2=\sqrt{s(s - a)(s - b)(s - c)}=\sqrt{31.5(31.5 - 17)(31.5 - 23)(31.5 - 23)}=\sqrt{31.5\times14.5\times8.5\times8.5}=\sqrt{32514.1875}=180.32\)

Step3: Calculate the area of the shaded region

Subtract the area of the smaller triangle from the area of the larger triangle.
\(A = A_1 - A_2=311.46-180.32 = 131.14\)

Answer:

\(131.14\)