Hello, Mhmh96!

Iyou made a sketch . . .assume

Triangle ABC has side lengths AB = 231, BC = 160, and AC = 281.

Point D is constructed on the opposite side of line AC as point B such that AD = 178 and CD = 153.

Compute the distance from B to the midpoint of segment AD.

The diagram seems to be something like this.

Draw segment AD.Code:A o 178 * * * D * * o * * * 231 * * * * 281 * * * * * 153 * * * * * * * ** B o * * * * * * * * * o C 160

We find that: .

. . Hence: .

Now the diagram looks like this:

LetCode:A o * ** * 1 * * 178 * * * * * * * * * D 231 * * o * 281 * * * * * * * 2* 153 * * * * ** B o * * * * * * * o C 160

Law of Cosines:

Hence: .

Finally, the diagram looks like this:

Draw segmentCode:A o * * 89 * * M 78 * - - o * : * 89 * : * D o - - : - - o * : * * : * 153 * : * 153 * : * * : * * : * B o - - o - - o C 80 N 80

We see that

is the hypotenuse of right triangle

. . Hence: .

Therefore: .