What is the slope of the line through (1, 9) and (–3, 16)?

For this case we have that by definition, the slope of a line is given by:

Where:

andare two points through which the line passes.

We have as data that:

Substituting we have:

Thus, the slope of the line is

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4