Geometry Question - Complex numbers & triangles

In summary, complex numbers are numbers with both real and imaginary parts that are used to represent points in geometry. They can be used to find the coordinates and area of triangles, prove geometric theorems, and provide more efficient solutions to some problems. However, they are not necessary for solving all geometry problems.
  • #1
MattL
11
0
OK, I've got this question to do:

Find complex numbers representing the vertices of a triangle ABC given
that the midpoints of the sides BC, CA, AB are represented by complex numbers
z_1, z_2, z_3 respectively.

Thing is, I don't know where I'm taking the origin to be; if I took it at A then I would just have A is O, B is 2*z_3, C is 2*z_2. Surely there's more to the question than that?
 
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  • #2
yeah, oops, just realized it's about similar triangle and it doesn't matter where the origin is...
 
  • #3


Yes, there is more to the question than simply finding the complex numbers representing the vertices of the triangle. In order to fully determine the triangle, we also need to know the location of the origin. This can be done by setting up a system of equations using the given information.

First, let's label the vertices as A(z_A), B(z_B), and C(z_C). We can set up the following equations based on the given information:

1) The midpoint of BC is z_1, so we know that z_1 = (z_B + z_C)/2
2) The midpoint of CA is z_2, so we know that z_2 = (z_C + z_A)/2
3) The midpoint of AB is z_3, so we know that z_3 = (z_A + z_B)/2

Now, we can rearrange these equations to solve for z_A, z_B, and z_C:

1) z_B = 2*z_1 - z_C
2) z_C = 2*z_2 - z_A
3) z_A = 2*z_3 - z_B

Substituting these values into each other, we can eliminate z_A, z_B, and z_C and solve for z_1, z_2, and z_3:

1) z_B = 2*z_1 - (2*z_2 - z_A)
2) z_C = 2*z_2 - (2*z_3 - z_B)
3) z_A = 2*z_3 - (2*z_1 - z_C)

Simplifying these equations, we get:

1) z_B = z_1 + z_A - 2*z_2
2) z_C = z_2 + z_B - 2*z_3
3) z_A = z_3 + z_C - 2*z_1

Now, we can choose any two of these equations and solve for z_A, z_B, and z_C. For example, if we choose equations 1 and 2, we get:

z_A = z_1 + z_2 - z_3
z_B = z_1 + z_2 - z_3
z_C = 2*z_2 - z_1

This means that the origin can be chosen at any point along the line z = z_1 +
 

Related to Geometry Question - Complex numbers & triangles

1. What are complex numbers and how are they used in geometry?

Complex numbers are numbers that have both a real part and an imaginary part, expressed as a + bi, where a and b are real numbers and i is the imaginary unit (√-1). In geometry, complex numbers are used to represent points on a two-dimensional plane, with the real part representing the x-coordinate and the imaginary part representing the y-coordinate. They are also useful in solving problems involving triangles and other geometric shapes.

2. How can complex numbers be used to find the coordinates of a triangle?

In a triangle, the three vertices can be represented as complex numbers. By using the distance formula and the properties of complex numbers, we can find the lengths of the sides of the triangle and use them to find the coordinates of the vertices. This can be useful in solving problems involving congruence, similarity, and other properties of triangles.

3. Can complex numbers be used to find the area of a triangle?

Yes, complex numbers can be used to find the area of a triangle. By representing the vertices of the triangle as complex numbers, we can use the shoelace formula to calculate the area. This formula involves taking the absolute value of the determinant of a matrix formed by the coordinates of the vertices, making use of the properties of complex numbers.

4. How can complex numbers be used to prove geometric theorems?

Complex numbers can be used to prove geometric theorems by representing geometric figures and their properties as complex numbers and using algebraic manipulation to prove the theorem. This is especially useful in proving theorems involving triangles, such as the Pythagorean theorem and the law of sines and cosines.

5. Are complex numbers necessary for solving geometry problems?

No, complex numbers are not necessary for solving all geometry problems. Many problems can be solved using only real numbers and basic geometric principles. However, complex numbers can provide a more elegant and efficient solution to some problems, and can also be useful in more advanced topics in geometry and other areas of mathematics.

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