PROVE THAT IF TWO LINES INTERSECT THEN THIER UNION IS IN THE SAME PLANE
if the two lines are not the same line, then pick 3 points:
the intersection, and one other point from each line.
Three points determine a single plane.
To prove that if two lines intersect, then their union is in the same plane, we can use Euclidean geometry. Here is a step-by-step proof:
Step 1: Assume that we have two lines, Line 1 and Line 2, that intersect at point P.
Step 2: Select any three arbitrary points on Line 1: A, B, and C.
Step 3: Line 1 consists of all the points that can be formed by joining A and B with line segment AB. Since AB is a segment on Line 1, all the points on Line 1 can be represented this way.
Step 4: Since Line 2 intersects Line 1 at point P, we can join P with any point on Line 1, such as point C, to form line segment CP.
Step 5: By the definition of a line, CP is a line that contains the points P and C. Therefore, CP is a line that intersects Line 1 at point C.
Step 6: Point C lies on both Line 1 and Line 2, which means that CP is a line that lies on both Line 1 and Line 2.
Step 7: By Steps 3 and 6, we have shown that Line 2 contains line segment CP, which has points from Line 1.
Step 8: By Steps 2 and 7, we have shown that Line 2 contains at least three points from Line 1.
Step 9: By the definition of a plane, any three non-collinear points determine a unique plane.
Step 10: Since Line 1 contains at least three points that determine a unique plane, and Line 2 contains at least three points that determine a unique plane, Line 1 and Line 2 are coplanar.
Step 11: Therefore, if two lines intersect, then their union is in the same plane.
This completes the proof.
To prove that if two lines intersect, then their union is in the same plane, we need to understand the concepts of lines, planes, and intersection.
1. Lines: A line is a straight path that extends indefinitely in both directions. In a three-dimensional space, a line can be represented by its direction and a point on the line.
2. Planes: A plane is a flat, two-dimensional surface that extends indefinitely in all directions. In a three-dimensional space, a plane can be defined by three non-collinear points or by a point on the plane and a normal vector.
3. Intersection: When two lines intersect, they meet at a common point. This point is shared by both lines and is called the point of intersection.
To prove that if two lines intersect, then their union is in the same plane, we can follow these steps:
Step 1: Assume we have two lines, Line A and Line B, in a three-dimensional space.
Step 2: Find the point of intersection between Line A and Line B. This can be done by finding the values of the variables that satisfy the equations of both lines simultaneously. The point of intersection will be the solution to the equations.
Step 3: Once we have the point of intersection, we can use it to define a plane. Let's call the point of intersection P.
Step 4: Choose two additional points on each line, not including the point of intersection. Call these points Q (from Line A) and R (from Line B).
Step 5: Use the three points P, Q, and R to define the plane that contains all three points. This can be done by finding the equation of the plane through the point P and having two direction vectors PQ and PR.
Step 6: Since the three points P, Q, and R lie on the same plane, we can conclude that the union of Line A and Line B lies entirely within this plane.
By following these steps, we have proven that if two lines intersect, then their union is in the same plane.