What is mass point theorem in geometry?

If two points balance, the product of the mass and distance from a line of balance of one point will equal the product of the mass and distance from the same line of balance of the other point.

What is mass point of a triangle?

Definition 1. A mass point is a pair (n, P), also written nP, consisting of a positive real number n, the mass, and a point P in the plane. Definition 2. We say two mass points coincide, nP = mQ, if and only if n = m and P = Q, i.e., they correspond to the same ordinary point with the same assigned mass.

How do you prove the Menelaus Theorem?

A standard proof is as follows: First, the sign of the left-hand side will be negative since either all three of the ratios are negative, the case where the line DEF misses the triangle (lower diagram), or one is negative and the other two are positive, the case where DEF crosses two sides of the triangle.

What is the center of mass of a triangle?

The centre of mass is also called the centre of gravity. The centre of mass of the triangle is the point at which the mass of the triangle will balance. To understand the “centre of mass” of a triangle, let us imagine balancing triangular cardboard on the pencil tip.

What is point mass give example?

The mass of an object, which is concentrated at a point is called ”point mass”. It has no internal structures like shape and size . Examples : (i) In the event of motion of Earth around the Sun , Earth can be treated as point mass . (ii) When stone is thrown in space, stone is considered as point mass .

How does Menelaus theorem differ from Ceva’s Theorem?

Ceva’s Theorem states that if the three Cevians of a triangle are concurrent then the previous statement holds. On the other hand, Menelaus’ Theorem states that if points D, E, and F on the sides BC, CA, and AB of triangle ABC are collinear, then the previous statement holds.

What is the converse of Menelaus Theorem?

Menelaus’ theorem relates ratios obtained by a line cutting the sides of a triangle. The converse of the theorem (i.e. three points on a triangle are collinear if and only if they satisfy certain criteria) is also true and is extremely powerful in proving that three points are collinear.

What is the difference between center of mass and geometric center?

In geometry, the centroid or geometric center of a figure is the average position of all the points included in the figure. On the other hand, the center of mass is the geometric point in which it would be necessary to place all its mass to have the same static moment that has the distributed mass.

What is point mass body?

Abstract. A point mass is an idealization of a real solid body. It possesses mass, but its demensions are assumed to be so small that its location can be sufficiently accurately defined by the position of a point.

What are mass points used for In geometry?

Mass points. Mass points is a technique in Euclidean geometry that can greatly simplify the proofs of many theorems concerning polygons, and is helpful in solving complex geometry problems involving lengths. In essence, it involves using a local coordinate system to identify points by the ratios into which they divide line segments.

How do you find the mass of a balancing line?

If two points are balanced, the point on the balancing line used to balance them has a mass of the sum of the masses of the two points. That is, if two points ( and ) have masses and , respectively, a third point between and which divides into the ratio will have mass .

How do you find the mass of a point dividing a line?

Additionally, the point dividing the line has a mass equal to the sum of the weights on either end of the line (like the fulcrum of a lever). The way to systematically assign weights to the points involves first choosing a point for the entire figure to balance around.