In triangle ABC, the median AM is perpendicular to the angle bisector BK. Find AB, is BC is 12 inches.
Let's start by drawing a diagram of triangle ABC:
```
B
/ \
A/_____ C
```
Now, let's label the given information:
```
B
/ \
M K
A/_____ C
BC = 12 inches
```
Since AM is a median, it divides side BC into two congruent segments. Let's label the length of each segment as X:
```
B
/ \
M K
A/______\ C
BC = 2X = 12 inches
```
Solving for X, we find that X = 12/2 = 6 inches.
Now, let's draw the angle bisector BK:
```
B
/ \
M K
A/___|___\ C
BC = 2X = 12 inches
```
Since AM is perpendicular to BK, angle AMK must be a right angle. Now, we can draw the right triangle AMK:
```
B
/ \
M K
A/___|___\ C
|
|
```
Now, let's solve for AB. Since AM is a median, it divides side BC into two congruent segments, so BM = 6 inches. Since angle AMK is a right angle, triangle AMK is a right triangle. Therefore, we can use the Pythagorean theorem to solve for AB:
AB^2 = AM^2 + BM^2
AB^2 = (2X)^2 + (6 inches)^2
AB^2 = (2*6 inches)^2 + 36 inches^2
AB^2 = 72 inches^2 + 36 inches^2
AB^2 = 108 inches^2
AB ≈ √108 inches
Using a calculator, we find that AB ≈ 10.39 inches.