In ∆ABC, the median AM (M ∈ BC) is perpendicular to the angle bisector BK (K ∈ AC).Find AB, if BC = 12 in.
AB = 6
It’s 6
To find AB, we need to use the properties of the medians and angle bisectors in a triangle.
First, let's draw a diagram to visualize the given information.
```
A
/ \
/ \
/ \
M /___B____\ C
\ /
\ /
\ /
\ /
K
```
In triangle ABC, we are given that BC = 12 inches, and BK is the angle bisector. Also, AM is the median of triangle ABC, and it is perpendicular to BK.
Now, let's use the properties of medians in a triangle. In a triangle, a median divides the opposite side into two equal parts. Therefore, we can say that BM = MC.
Now, let's use the properties of angle bisectors in a triangle. An angle bisector divides the opposite side into two segments that are proportional to the adjacent sides. In this case, we have BK as the angle bisector, and it divides AC into two segments, AK and KC.
Since AM is perpendicular to BK, it forms a right angle. This means that triangle ABM is a right triangle. We can use the Pythagorean theorem to find the length of AB.
According to the Pythagorean theorem, the square of the length of the hypotenuse (AB) is equal to the sum of the squares of the other two sides (AM and BM).
So, let AB = x. We have:
AB^2 = AM^2 + BM^2
Since BM = MC (due to the properties of medians), we can rewrite the equation as:
AB^2 = AM^2 + (BC/2)^2
Now, substitute the known values:
AB^2 = AM^2 + (12/2)^2
AB^2 = AM^2 + 6^2
AB^2 = AM^2 + 36
Since we don't have the length of AM given, we cannot directly solve for AB. We need more information about the median AM to find AB accurately.