A 12 cm segment AB is divided by points M and N into three equal parts. The projection of the segment MN

A 12 cm segment AB is divided by points M and N into three equal parts. The projection of the segment MN onto the ray AC is 2 cm.Find the cosine of the angle BAC

By the condition of the problem:

| AM | = | MN | = | NB | = 4;

Let K; L; C – projections of points M; N and B per AC beam. Then:

From a right-angled triangle NAL: | AL | = | AN | * cos (BAC);

From the right-angled triangle MAK: | AK | = | AM | * cos (BAC);

Subtracting the second from the first equality, we get:

| AL | – | AK | = | AN | * cos (BAC) – | AM | * cos (BAC) = (| AN | – | AM |) * cos (BAC) = | MN | * cos (BAC);

Considering that by the condition of the problem:

| AL | – | AK | = | KL | = 2;

we get:

2 = 4 * cos (BAC);

cos (BAC) = 1/2;

Answer: The cosine of the angle BAC is 1/2.