Finding Coordinates of last Triangle Vertex

In summary, the conversation is about finding the last vertex coordinates of a triangle given two vertices and the angles at those vertices. The speaker has tried using equations based on the lengths of the sides and the cos dot product rule, but has only ended up with one equation for two unknowns. They are then advised to use the sine law to find the distance between the two given vertices and the angle at the third vertex. By setting up equations using the distance and angles, the speaker is able to find the coordinates of the third vertex using the simultaneous equations method. However, they encountered an issue of having two possible solutions, both with the same angle between the vectors.
  • #1
monch
3
0
Hi,
I am trying to find the last vertex coordinates of a triangle given that

Vertex 1 = (2,10)
Vertex 2 = (3,6)

Angle at Vertex 1 = 75.9638 degrees
Angle at Vertex 2 = 70.3462 degrees.


I have tried using the equations based on the length of each side, as well as using the cos dot product rule to try to find x,y but I cannot seem to .

I always end up with the same equations using both methods, hence only giving me 1 equation for 2 unknowns.
 
Physics news on Phys.org
  • #2
You can use the sine law. The distance between vertices 1 and 2 is [itex]\sqrt{(2- 3)^2+ (10- 6)^2}[/itex][itex]= \sqrt{1+ 16}= \sqrt{17}[/itex].
Further, the angles must add to 180 degrees so the angle at vertex 3 is [itex]180- 75.9638- 70.3462= 33.69[/itex] degrees.
So the length, r1, of the side from vertex 1 to vertex 3, the side opposite vertex 2, is given by [itex]r1/sin(70.3462)= \sqrt{17}/sin(33.69)[/itex]. Write the equation of the circle with center at vertex 1 with that radius.
And the length, r2, of the side from vertex 2 to vertex 3 the side opposite vertex 1, is given by [itex]r2/sin(75.9638)= \sqrt{17}/sin(33.69)[/itex]. Write the equation of the circle with center at vertex 2 with that radius.

Find the points where those two circles intersect by solving the equations simultaneously. There are two solutions, one on either side of the line from vertex 1 to vertex 2.
 
  • #3
Thanks for the reply HallsofIvy,

I end up with only 2 complex roots for some reason when i use these 2 equations

(x-3)^2 + (y - 6)^2 = 52 =(r2^2)
(x-2)^2 + (y-10)^2 = 49 = (r1^2)

i got 10x^2 - 284x +2652 = 0 after substitution.

the answer i am suppose to get for the last coordinate is (9,10)
 
  • #4
oops my mistske, i did some calculation errors.

hmm is there anyway to pick the right coordinates when i end up with 2 solutions?

Both will have the same angle between the vectors , so there's no way to tell which is the coordinate I am looking for?
 

FAQ: Finding Coordinates of last Triangle Vertex

1. How do I find the coordinates of the last vertex in a triangle?

The coordinates of the last vertex in a triangle can be found using the distance formula and the properties of triangles. First, determine the length of each side of the triangle using the given coordinates of the other two vertices. Then, use the distance formula to find the distance between the known vertices and the unknown vertex. Finally, use the properties of triangles (such as the Pythagorean theorem) to calculate the coordinates of the last vertex.

2. Can I use the slope formula to find the coordinates of the last vertex in a triangle?

Yes, the slope formula can be used to find the coordinates of the last vertex in a triangle. The slope formula, also known as the rise over run formula, can be used to determine the slope of a line. By finding the slope of the two sides of the triangle that are known, you can then use the slope-intercept form of a line to find the coordinates of the last vertex.

3. Do I need to know the angle measurements of a triangle to find the coordinates of the last vertex?

No, you do not need to know the angle measurements of a triangle to find the coordinates of the last vertex. The distance formula and properties of triangles are sufficient in finding the coordinates. However, knowing the angle measurements can make the calculation process easier and more accurate.

4. What if the triangle is not a right triangle, can I still use the Pythagorean theorem to find the coordinates of the last vertex?

Yes, the Pythagorean theorem can be used to find the coordinates of the last vertex in any type of triangle, whether it is a right triangle or not. This is because the Pythagorean theorem relates the lengths of the sides of a right triangle, but it can also be applied to other types of triangles by using the Law of Cosines or the Law of Sines.

5. Are there any other methods or formulas for finding the coordinates of the last vertex in a triangle?

Yes, there are other methods and formulas for finding the coordinates of the last vertex in a triangle. Some examples include using the midpoint formula and the distance formula, as well as using trigonometric functions such as sine, cosine, and tangent. However, these methods may require additional information or calculations and may not be as straightforward as using the distance formula and properties of triangles.

Similar threads

Replies
3
Views
1K
Replies
2
Views
2K
Replies
2
Views
1K
Replies
1
Views
985
Replies
1
Views
1K
Replies
6
Views
794
Back
Top