ghostfirefox said:Point D divides side AC, of triangle ABC, so that |AD|: |DC| = 1:2. Prove that vectors [tex]\vec{BD}[/tex] = 2/3 [tex]\vec{BA}[/tex] + 1/3 [tex]\vec{BC}[/tex].
HallsofIvy said:You titled this "Analytic geometry proof ..." so I would set up a coordinate system so that the origin is at the vertex "B" of the triangle and the x-axis lies along on side BC. Then B= (0, 0), A= (x, y), and C= (z, 0) for some numbers x, y, and z. Since D lies on AC such that "|AD|:|DC|= 1:2", D= ((2x+ z)/3, 2y/3). Now, what are the vectors [tex]\vec{BA}[/tex], [tex]\vec{BC}[/tex]. and [tex]\vec{BD}[/tex]?
Analytic geometry is a branch of mathematics that combines algebra and geometry to study geometric shapes and their properties using coordinate systems and equations.
A proof in analytic geometry is a logical argument that uses established mathematical principles and equations to show that a given statement or theorem is true for all cases.
To prove a triangle in analytic geometry, you can use various methods such as the distance formula, slope formula, and the Pythagorean theorem to establish the relationships between the coordinates of the triangle's vertices and show that the given statement or theorem holds true.
The key elements of a triangle proof in analytic geometry include the given information, the diagram or figure of the triangle, the coordinates of the triangle's vertices, the equations used to establish relationships between the coordinates, and the logical steps taken to arrive at the proof.
Proofs are important in analytic geometry because they provide a rigorous and systematic way to verify the validity of geometric statements and theorems. They also help to deepen our understanding of geometric concepts and develop problem-solving skills.