The acute angle between two straight lines is given by:

where and are the gradients of the lines.

### Examples

1. Find the acute angle between the lines and .

### Show Answer

2. If the angle between the lines and is , find two possile values of *m*, correct to 1 decimal place.

### Show Answer

There are two possibilities:

...OR...

## A Point Between an Interval

The coordinates of a point *P* that divides the interval between points and in the ratio *m*:*n* respectively are given by:

If the point *P* divides the interval externally in the ratio *m:n*, then the ratio is negative and *P* lies outside *AB*.

### Examples

1. Divide *AB* into the ratio 3:4 where *A* is (6 -2) and *B* is (-7, 5).

### Show Answer

2. If *A* is (-2, -1) and *B* is (1, 5), find the coordinates of the point *P* that divides *AB* externally in the ratio 2:5.

### Show Answer

Let the ratio be 2:-5 (you can also use -2:5, with the same result).

### Extension: Angle Between 2 Curves

To measure the angle between two curves, measure the angle between the tangents to the curves at that point.

where and are the gradients of the tangents to the curves at the point of intersection.

For example...

Find the acute angle formed at the intersection of the curves and .