In an oblique triangle ABC, A=30 degree, and the perpendicular from C to AB is 12 inches long. Find the length of AB.

tan 30 = 12/AB

Actually, you don't say that B is at the foot of the perpendicular. All we know is that the altitude from C to AB is 12. There is nothing to say how long AB is.

In fact, we are told that ABC is an oblique triangle, indicating that it is fact not a right triangle.

To find the length of side AB in an oblique triangle, we can use the law of sines. The law of sines states that the ratio of a side length to the sine of its opposite angle is a constant.

In this case, we have angle A = 30 degrees and side opposite A is AC, which is the perpendicular from C to AB. Let's call the length of side AB as x.

Using the law of sines, we can set up the following equation:

AC / sin(A) = AB / sin(C)

Given that angle A = 30 degrees, we can substitute the values:
12 / sin(30) = x / sin(C)

Now, we need to find angle C. Since we know that the sum of angles in a triangle is 180 degrees, we can find angle C using the equation: C = 180 - A - B

For an oblique triangle, B can be calculated using: B = 180 - A - C

Substituting A = 30 degrees, we get: B = 180 - 30 - C = 150 - C degrees

Since we know that angle B is the angle opposite side AB, we can rewrite the equation as:
sin(B) = sin(150 - C)

Finally, we substitute this equation back into the equation using the law of sines to solve for x:

12 / sin(30) = x / sin(C) * sin(150 - C)

Solving this equation will give us the length of side AB, x.