DIVISIÓN DE UN SEGMENTO EN UNA RAZÓN DADA

Summary

This lesson explains the concept of dividing a segment into a given ratio and how to calculate the coordinates of the point that divides it. The process is demonstrated using Cartesian coordinates and formulas for both x and y components. Additionally, the video walks through the calculation of the centroid (barycenter) of a triangle, which is the intersection of its three medians. The centroid divides each median in a 2:1 ratio. The example given uses the coordinates of a triangle to illustrate how to find the centroid, providing a clear application of these geometric principles.

Takeaways

• 😀 The script discusses the process of dividing a segment into a given ratio using coordinates in the Cartesian plane.
• 😀 The segment is labeled 'AB', and the coordinates of the two endpoints, A and B, are used in the calculations.
• 😀 A point 'P' divides the segment into two parts, with one part being 'AP' and the other part 'PB'.
• 😀 The ratio between the segments 'AP' and 'PB' is a crucial factor in determining the coordinates of point 'P'.
• 😀 To find the coordinates of point 'P', the x-coordinate is calculated using the formula: (x1 + r * x2) / (1 + r), where 'r' is the ratio.
• 😀 A similar formula is used to calculate the y-coordinate of point 'P', considering the ratio and the coordinates of both endpoints.
• 😀 The method can also be applied in reverse, with the ratio 'PB' over 'AP', but the formula adjustments must be carefully considered.
• 😀 An example involving a triangle is used to apply the concepts, specifically calculating the centroid (baricenter) of a triangle.
• 😀 The centroid of a triangle is the point where its three medians intersect, and it divides each median in a 2:1 ratio.
• 😀 The calculation of the centroid involves finding the midpoint of a segment, determining the ratio of the median, and using the coordinates of the triangle's vertices to calculate the centroid's position.
• 😀 The final result of the example shows that the centroid of the triangle has coordinates (3, 3).

Q & A

What is the main topic discussed in the video?

The main topic of the video is how to divide a segment in a given ratio and how to find the coordinates of the point that divides the segment using the concept of Cartesian coordinates.

What is the purpose of knowing the coordinates of the endpoints of a segment?

Knowing the coordinates of the endpoints of a segment is essential because it helps in calculating the coordinates of any point that divides the segment in a given ratio.

How are the coordinates of point P calculated when a segment is divided in a ratio?

The coordinates of point P are calculated by using a formula where the abscissa of point P is found by summing the abscissas of the endpoints weighted by the given ratio, and dividing by the sum of 1 and the ratio.

What is the significance of the direction in which the ratio is calculated?

The direction in which the ratio is calculated is important because it determines the order in which the coordinates of the endpoints are used in the formula. If the ratio is taken from point A to point B, the formula uses the abscissas and ordinates starting from point A.

How do you calculate the ordinate of point P when the segment is divided?

The ordinate of point P is calculated by using a similar process to the abscissa calculation. You add the ordinates of the endpoints weighted by the ratio and then divide by the sum of 1 and the ratio.

What is the baricentro (centroid) of a triangle?

The baricentro (centroid) of a triangle is the point where the three medians of the triangle intersect. It divides each median into two parts, with the ratio of 2:1, where the longer segment is closer to the vertex.

What are the steps to find the centroid of a triangle?

To find the centroid, you first identify the medians of the triangle. Then, calculate the point of intersection of these medians, or use the known property that the centroid divides each median in a 2:1 ratio.

How is the midpoint of a segment used in finding the centroid of a triangle?

The midpoint of a segment is crucial in calculating the medians. In this video, the midpoint of segment AC is used to help construct the median, which will lead to finding the centroid.

What formula is used to calculate the coordinates of the centroid?

The coordinates of the centroid are found by averaging the coordinates of the three vertices of the triangle. The formula for the centroid G is Gx = (x1 + x2 + x3) / 3, and Gy = (y1 + y2 + y3) / 3.

What is the role of the 2:1 ratio in the calculation of the centroid?

The 2:1 ratio in the centroid calculation indicates that the centroid divides each median into two segments: the segment closer to the vertex is twice as long as the segment closer to the midpoint of the opposite side.