Problem 49 - Thomas Calculus

Question

Medians of a triangle Suppose that \(A, B,\) and \(C\) are the corner points of the thin triangular plate of constant density shown here. a. Find the vector from \(C\) to the midpoint \(M\) of side \(A B\) . b. Find the vector from \(C\) to the point that lies two-thirds of the way from \(C\) to \(M\) on the median \(C M .\) c. Find the coordinates of the point in which the medians of \(\Delta A B C\) intersect. According to Exercise \(29,\) Section \(6.4,\) this point is the plate's center of mass.

Step-by-Step Solution

Answer

a) Vector \( \overrightarrow{CM} \) is computed from given coordinates. b) The point is \( C + \frac{2}{3}(\overrightarrow{CM}). \) c) Centroid \( G = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right). \)

Step 1: Determine Midpoint M of Side AB

To find the midpoint \( M \) of side \( AB \), use the midpoint formula for each coordinate: \[ M = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) \]Substitute the coordinates of \( A \) and \( B \) into the formula.

Step 2: Calculate Vector from C to M

With the coordinates of \( M \) found in Step 1, calculate the vector \( \overrightarrow{CM} \) using vector subtraction: \[ \overrightarrow{CM} = M - C = \left( \frac{x_A + x_B}{2} - x_C, \frac{y_A + y_B}{2} - y_C \right) \]

Step 3: Point Two-Thirds of the Way from C to M

To find the point two-thirds of the way from \( C \) to \( M \) on the median \( CM \), use the formula: \[ P = C + \frac{2}{3}(\overrightarrow{CM}) \]Here, find the new coordinates by scaling \( \overrightarrow{CM} \) by \( \frac{2}{3} \) and adding to \( C \).

Step 4: Find the Centroid of Triangle ABC

The centroid \( G \) of \( \Delta ABC \), where the medians intersect, is the average of the coordinates of \( A, B, \) and \( C \): \[ G = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right) \]

Key Concepts

Median of a Triangle

A median in a triangle is a crucial line segment that connects one of the triangle’s vertices to the midpoint of the opposite side. Every triangle has three medians, and these lines play an important role in defining the geometric properties of the triangle. To find the midpoint of a side, like side \(AB\) in a triangle, use the midpoint formula: \[ M = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) \]. Here \(M\) becomes the midpoint of side \(AB\), making it a key component in various calculations involving medians.
Each median divides the triangle into two smaller triangles of equal area. Collectively, the three medians intersect at a single point known as the centroid.
This centroid divides each median into two segments, with the portion between the vertex and the centroid being twice as long as the portion between the centroid and the midpoint of the side.

Vector Calculation

In vector calculation, understanding how to manipulate these mathematical objects is essential. Vectors help us describe various physical quantities such as displacement and force. To calculate a vector from one point to another, say \(C\) to \(M\), use vector subtraction: \[ \overrightarrow{CM} = M - C = \left( \frac{x_A + x_B}{2} - x_C, \frac{y_A + y_B}{2} - y_C \right) \].
From the vector \(\overrightarrow{CM}\), you can perform further calculations, such as scaling by a factor. This involves multiplying each component of the vector by the given scalar.

• This property is particularly useful when finding specific points on a median or path, such as a point that is two-thirds of the distance from the starting point \(C\) to the midpoint \(M\).

By dissecting vector operations, students can thoroughly understand geometric movements and transformations within the triangle.

Triangle Geometry

Triangle geometry is a fascinating branch of math that delves into the properties and relations of triangles. Within a triangle, there are several significant points and lines, with medians being a primary example.

• Each triangle has precisely three medians.
• These medians intersect at a single unique point called the centroid, which acts as the "center" of the triangle in some respects.
  

This centroid plays an important role in both geometry and physics, essentially balancing the triangle evenly.
In more complex geometric problems, understanding the properties and behaviors of medians provides insights into optimizing constructions and problem-solving strategies. Mastery in triangle geometry involves visualizing and analyzing these components effectively.

Center of Mass

The center of mass, often interchangeable with the term centroid in the context of triangles, is the point where the medians of the triangle intersect. It represents an average position of mass distribution if the triangle is of uniform density.

• In mathematical terms, for a triangle \( \Delta ABC \), the coordinates of the center of mass or centroid \( G \) are given by: \[ G = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right) \].
• The centroid divides each median into a 2:1 ratio.

This meaningful point has an important application in physics and engineering, illustrating how a body would balance on a single support point.
Identifying the center of mass ensures a better understanding of object stability and balance, crucial in designing structures or studying motion.