Problem Analysis #30

[THOUGHT] Given information allows us to use the Law of Cosines to find side BC of triangle ABC. The Law of Cosines states that for any triangle with sides a, b, c and angle C opposite side c, c² = a² + b² - 2ab * cos(C). Here we know a=5 (AB), b=6 (AC), and cos(∠BAC)=-3/5. Let's use the Law of Cosines to find c (BC).

[PYTHON]

import math

# Given values
AB = 5
AC = 6
cos_BAC = -3 / 5

# Using Law of Cosines to calculate BC
BC_squared = AB**2 + AC**2 - 2*AB*AC*cos_BAC
BC = math.sqrt(BC_squared)

# Calculate area of triangle ABC using Heron's formula
s = (1/2) * (AB + AC + BC)
Area = math.sqrt(s*(s-AB)*(s-AC)*(s-BC))

print("The area of triangle ABC is:", Area)

Running the code gives: The area of triangle ABC is: 12.0

[THOUGHT] Hence, the area of triangle ABC is 12 square units.

$\boxed{12}$

[THOUGHT] Based on the Python output where the area of triangle ABC is approximately 12.0 square units, we can confidently conclude that:

$\boxed{12}$

is the correct answer for the area of triangle ABC. This value matches with our previous calculation using Heron's formula and gives us a precise measurement for the triangle's area based on the provided side lengths and angle information.