i need help with is question about the angle between two intersecting lines, it goes:
the points (0,2), (2,3), (4,0) and (2,-1) are vertices of a quadrilateral. find the tangent of the acute angle between its diagonals. the answer is 2 but idk how they got that. and also wats it mean by find the tangent??

Question

i need help with is question about the angle between two intersecting lines, it goes:
the points (0,2), (2,3), (4,0) and (2,-1) are vertices of a quadrilateral. find the tangent of the acute angle between its diagonals. the answer is 2 but idk how they got that. and also wats it mean by find the tangent??

anonymous · Answer

Can you use vectors?  or straight geometry?

anonymous · Answer

yea i think u can

anonymous · Answer

Then, find the vector u that goes between (2,3) and (2,-1), and the vector v that goes between (0,2) and (4,0), then find the dot product

anonymous · Answer

Do you know how to do this?

anonymous · Answer

no :/

anonymous · Answer

The vector u goes between ((2,3) and (2,-1) so it is <0,-4>
Do you see that?

anonymous · Answer

The vector v goes between (0,2) and (4,0) so it is <4,-2>
The angle brackets mean vectors, the parentheses are points.

anonymous · Answer

The dot product is <0,-4> • <4,-2> Do you know how to compute that?

anonymous · Answer

so u find half why between them?

anonymous · Answer

way**

anonymous · Answer

That's the geometry way.  We can do that if you are more comfortable.  The geometry method would find the point halfway, then use the law of cosines to find the angle.  The vector method is a bit easier, it uses the definition of the dot product, which I'm getting to.  But we can do it either way.

anonymous · Answer

which ever 1 is easier to remember 
but i still dont understand this dot thing

anonymous · Answer

I like vectors better so we'll try it.  OK?

anonymous · Answer

Do you understand how I got the vectors?

anonymous · Answer

not really

anonymous · Answer

Let's do geometry.  I think I see a short cut.

anonymous · Answer

ok :)

anonymous · Answer

What you need to do is find the point where the diagonals cross.

anonymous · Answer

Sorry @bishay187 but I gotta go to work.  Why not repost or call for help.  OK?

anonymous · Answer

alright thanks anyway

anonymous · Answer

I found time to continue, hope you come back and see this.

I plotted your points.  Should have done that the very first thing.  Notice that the quadrilateral is a parallelogram.  The diagonals cross half-way between the opposing vertices, at (2,1), labeled B in the figure.

Now draw the altitude between (2,0) and (2,2), labeled AC.

The angle for which you want the tangent is angle ABC.  Can you compute the tangent from the figure?