In analytic geometry: The topic regarding the angle between two lines...?

Question

Lemme give an example. The points W(-2,7), J(3,5), and M(-6,8) are vertices of a triangle. Find the interior angles.

-> And my QUESTION is:
Right we have to try to plot the points first in the Cartesian plane, before we graph on how the triangle would look like? Eh... Our teacher insists that the positions of the points don't matter in arriving at the right answer, but it does matter... =D

WHAT I MEAN IN MY QUESTION IS... WHAT IF WHEN YOU PLOT THE POINTS, POINT J SHOULD BE BESIDE POINT W, AND POINT M ON TOP OF THEM BOTH? AND WHAT HAPPENS IF WE CHANGE THIS? TAKE NOTE THAT THIS IS A TRIANGLE. IF WE CHANGE HOW IT REALLY WOULD LOOK LIKE IN THE COORDINATE PLANE, HOW CAN WE ARRIVE AT THE SAME ANSWER? WE'VE TRIED IT AND GOT NEGATIVE INTERIOR ANGLES OF A TRIANGLE, YET OUR TEACHER SAYS THAT THE ARBITRARY DRAWING OF A TRIANGLE DOESN'T MATTER, EVEN THOUGH IT WOULD BE REALLY DIFFERENT FROM WHAT IT WOULD APPEAR IN THE COORDINATE PLANE. AND SO... CAN WE STILL SAY THAT IT'S RIGHT?

I want an answer to my question, not the answer to the problem. Hope you'd remember this.

Answer 1

I think I know what your teacher means by saying the positions of the points don't matter. If you move your points W, J, M three units to the right and two up, you get different co-ordinates, and the triangle is in a different position. However, as the points are in the same position relative to each other, the angles and other details of the triangle are the same as before. Only its position has changed.

You need the find the gradient of each side of the triangle. The gradient equals the tangent of the angle between the side and the x-axis.
If your first two sides have gradients m1, m2, then the angle t between them satisfies the relationship:
tan(t) = (m1 - m2) / (1 + m1m2)
based on the formula:
tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B)).

Then do the same thing for another pair of sides, and once you have two angles, you can subtract the total from 180deg. to get the third.

Answer 2

Your teacher is absolutely correct --- you don't NEED to plot these points first, it will simply be a visualization aid.

Given the points, you can easily find the lengths of each side of the triangle. After that you can find the interior angles, such as:

tan(ang. WJ) = (-2/5)
so (ang. WJ) = arctan(-2/5)

Answer 3

Consider the angle created at the intersection of W & J.
The tangent of that angle is the difference of the y-values divided by the difference of the x-values of W and J.
tan(ang. WJ) = (-2/5)
so (ang. WJ) = arctan(-2/5)

Repeat this for each angle, and it's a breeze.

Answer 4

the first question, the line is 1/4(I dont know what 7 untis to the left thing is sry :/)

Answer 5

Use the formula
v1•v2 = |v1||v2| cos(angle)

JW=(-5,2)
JM=(-9,3)
|JW|=sqrt(29)
|JM|=sqrt(90)

JW.JM=45+6=51

51=sqrt(29).sqrt(90) . Cos J

Cos J=51/sqrt(2610)

J=use your calculator.
Do the same for other angles.