Question

choose 1 answer:
a 38.6
b 43.5
c 75
d 87

Explanation:

Step1: Calculate the length of \(AB\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(A(1,8)\) and \(B(6,6)\), \(x_1 = 1,y_1 = 8,x_2=6,y_2 = 6\).
\(AB=\sqrt{(6 - 1)^2+(6 - 8)^2}=\sqrt{25 + 4}=\sqrt{29}\)

Step2: Calculate the length of \(AC\)

For \(A(1,8)\) and \(C(-5,-7)\), \(x_1 = 1,y_1 = 8,x_2=-5,y_2=-7\).
\(AC=\sqrt{(-5 - 1)^2+(-7 - 8)^2}=\sqrt{36+225}=\sqrt{261}\)

Step3: Calculate the area of right - triangle \(ABC\)

The area formula for a right - triangle is \(S=\frac{1}{2}\times AB\times AC\). Substitute \(AB = \sqrt{29}\) and \(AC=\sqrt{261}\) into the formula.
\(S=\frac{1}{2}\times\sqrt{29}\times\sqrt{261}=\frac{1}{2}\times\sqrt{29\times9\times29}=\frac{1}{2}\times3\times29 = 43.5\)

Answer:

B. \(43.5\)