What are Collinear Points in Geometry?
Geometry is considered to be one of the most important subjects in school. While it may seem confusing at first, there are a few simple things that you can do in order to better understand and remember these concepts. In this article, let us understand and learn about collinear points.
Collinear points have three or more points aligned on a straight line. If we take some real-life example of collinearity, it would be like people standing in a queue. Hence, we can say collinear points are all situated on the very same line itself. In cartesian geometry, it is said that if the area formed by the given points is zero, then the points can be said to be collinear. For more facts on collinear points, click here.
Collinear points are important for geometry because it helps us identify the angles of any two lines when there are multiple points between them. It allows us to see if two lines intersect at all, what is the exact point of intersection which is called the point of concurrency. Collinear points can be said to be aligned, or they can be said to be in a row.
Geometry starts with the basics of points, moving on to lines and then the shapes formed by these points and lines, or their intersection. In Euclidean geometry, Euclid defined a line as “breadthless length” and a straight line as “a line that lies equally with the points on itself” when he first formalized geometry.
A line is often described as a group of points following a linear connection when some other basic notion is left undefined in modern geometry; however, it is occasionally described as an undefined object with qualities supplied by axioms. A collineation is a mapping of geometry to itself that transfers lines to lines and retains the collinearity attribute. The linear mappings are known as homographies in the world of projective geometry. When seen as geometric maps, linear mappings (or linear functions) of vector spaces link lines to lines; that is, they map collinear point sets to collinear point sets and hence are collinear.
Within any geometric figures, there can also be collinear points, like for example, the area centroid, the vertex centroid, and the intersection of the diagonals are all collinear in a cyclic quadrilateral. The midpoints of the legs are collinear with the incenter in a tangential trapezoid. Collinearity exists between the incenter of a triangle, the midpoint of an altitude, and the point of contact of the corresponding side with the excircle relative to that side.
At collinear places, the lines connecting the foot of the elevations intersect the opposing sides. The center, two foci, and two vertices of a hyperbola are all collinear. The halfway between a tetrahedron’s Monge point and circumcenter is the centroid.
The concept of collinearity can further be practiced with the help of math worksheets which are easily available online. The students can visit the Cuemath website to download them; also, they will get access to the answer keys as well. Solving math worksheets related to geometry will help the students explore these topics in much detail as well as with real-world examples. The questions in these worksheets are presented from easy to complex range depending on the intellect of the student so that they can navigate through it accordingly.
Practicing worksheets will increase mental math abilities, which may help you do faster calculations and study easier in higher grades, as well as in real-life situations when you need to do calculations. The tasks in the worksheets will help you relate what you have encountered in the real world to the concept of geometry by using real-life explanations and difficulties.