Point N is on hypotenuse BC of triangle ABC such than angle CAN is 45 degrees.

If AC=8 and AB=6, find AN.

To find the length of AN, we need to use the trigonometric ratios in a right-angled triangle.

First, let's draw triangle ABC with the given information. The triangle will have a right angle at angle C.

A
|\
| \
//\
// \
// \
B---------C

Now, let's label the points:
- A is the angle opposite to side BC (hypotenuse).
- B is the angle opposite to side AC (adjacent).
- C is the right angle.
- N is a point on BC.

Given information:
- AC = 8 units
- AB = 6 units
- Angle CAN = 45 degrees

To find AN, we can use the sine or cosine ratio.

Using the sine ratio:
sin(angle) = opposite/hypotenuse

In this case, we have angle CAN, where AN is the opposite side and AC is the hypotenuse.

sin(CAN) = AN / AC

We can rearrange this equation to solve for AN:

AN = sin(CAN) * AC

Now, let's substitute the known values:

AN = sin(45 degrees) * 8

To find the value of sin(45 degrees), you can use a trigonometric table, calculator, or you can approximate it by knowing that sin(45 degrees) = 1/sqrt(2) = 0.7071.

AN = 0.7071 * 8

AN ≈ 5.657 units

Therefore, the length of AN is approximately 5.657 units.