Section and Mid-Point Formula - Class 10 MCQ

To find the coordinates of a point that trisects the line segment, you would apply the section formula twice: once for the ratio 1:2 and again for the ratio 2:1. This allows you to find both points of trisection accurately. Trisection is a useful technique in various applications, from engineering to art.

The midpoint formula is calculated by averaging the x-coordinates and y-coordinates of points A and B. Thus, the midpoint is ((2 + 6)/2, (1 + 5)/2) = (4, 3). Finding midpoints is fundamental in geometry, as it helps in determining central locations in various geometric constructions.

To find the coordinates of point P that divides the line segment, we use the Section Formula. Plugging in the coordinates of points A and B along with the specified ratio gives us P = (32/7, -19/7). This method illustrates how ratios affect the position of points within geometric figures, enabling deeper understanding of coordinate relationships.

The centroid of a triangle is found using the formula for the average of the vertices' coordinates: Centroid = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3). Calculating this for the given points gives us ((1 + 4 + 7)/3, (2 + 6 + 2)/3) = (4, 3). The centroid is a crucial point that represents the triangle's center of mass.

The perpendicular bisector of a line segment divides the segment into two equal parts while intersecting it at a right angle. This concept is significant in geometry, especially in constructions and proofs involving triangles and circles, as it helps in defining symmetry and balance.

The coordinates of the centroid are determined by averaging the x-coordinates and the y-coordinates of the triangle's vertices. This relationship highlights the centroid's role as a balance point of the triangle, providing crucial insights into the triangle's geometric properties and symmetry.

To find the y-coordinate of the midpoint, you calculate (y1 + y2)/2 = (-1 + 7)/2 = 3. Knowing how to find midpoints is fundamental for problems involving symmetry and balancing in geometric figures.

The section formula is employed to find the coordinates of a point dividing a line segment in the ratio k:1. This formula is crucial in coordinate geometry, enabling the precise location of points along a line segment based on proportional divisions, which is often required in geometric constructions and analyses.

Points of trisection divide a line segment into three equal parts, resulting in one point dividing it in the ratio 1:2 and the other in the ratio 2:1. Understanding trisection is important in geometry for constructing equal segments, which is often needed in more complex geometric designs.

The Section Formula is used to find the coordinates of a point that divides a line segment into a specific ratio. This is essential in coordinate geometry, as it allows for precise location of points based on proportional distances along a line segment. For instance, if a line segment is divided in the ratio of 2:5, the formula provides the exact coordinates of the dividing point based on the coordinates of the endpoints.