Problem 53 - University Calculus: Early Transcendentals

Question

Medians of a triangle \(\quad\) Suppose that \(A, B,\) and \(C\) are the comer 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 \(19 .\) Section \(6.6,\) this point is the plate's center of mass. (See the figure.) Figure cannot copy

Step-by-Step Solution

Answer

The vectors are \( \vec{CM} = (x_m - x_3, y_m - y_3) \), and the centroid is \( G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \).

Step 1: Calculate the Midpoint of AB

To solve part (a), first find the midpoint \( M \) of side \( AB \). If \( A(x_1, y_1) \) and \( B(x_2, y_2) \) are the coordinates of points \( A \) and \( B \), the midpoint is found using the formula:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]

Step 2: Find Vector from C to M

Now, find the vector from \( C(x_3, y_3) \) to the midpoint \( M(x_m, y_m) \) using the formula:\[\vec{CM} = (x_m - x_3, y_m - y_3)\]

Step 3: Identify the Point 2/3 of the Way Along CM

For part (b), find the point that is two-thirds of the way from \( C \) to \( M \) on the median using the vector \( \vec{CM} \). This point is given by:\[P = C + \frac{2}{3} \vec{CM} = (x_3, y_3) + \frac{2}{3} \cdot (x_m - x_3, y_m - y_3)\]Simplifying, we get:\[P = \left( x_3 + \frac{2}{3}(x_m - x_3), y_3 + \frac{2}{3}(y_m - y_3) \right)\]

Step 4: Find Intersection of the Medians

For part (c), calculate the intersection of the medians, which is the centroid \( G \). The centroid is found by calculating the average of the coordinates of the three vertices:\[G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)\]

Key Concepts

Understanding Midpoints in Triangles

When working with triangles, the midpoint of one of its sides is a crucial point to know. It cuts the line segment connecting two vertices into equal halves. In a triangle with vertices labeled as \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \), the midpoint \( M \) of the side connecting vertices \( A \) and \( B \) can be easily determined.
The formula to find the midpoint \( M \) is:

• \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

This formula calculates the average of the x-coordinates and the y-coordinates of the two endpoints \( A \) and \( B \).
Knowing midpoints helps in solving more complex problems, such as finding the medians or the centroid of the triangle.

Role of Medians in Triangles

Medians of a triangle are special line segments. Each median connects a triangle’s vertex to the midpoint of the opposite side. Understanding medians is key to determining the centroid, or balancing point, of a triangle. In a triangle with vertices \( A, B, \) and \( C \), the median from vertex \( C \) to the midpoint \( M \) of \( AB \) is essential.

This median can be represented as the vector \( \vec{CM} \):

• Calculate \( \vec{CM} \) by \( \vec{CM} = (x_m - x_3, y_m - y_3) \)
  where \( M(x_m, y_m) \) is the midpoint of side \( AB \) and \( C(x_3, y_3) \).

Medians are crucial because they intersect at the triangle's centroid. Each median divides the triangle into two sections of equal area, emphasizing the medians' significance in geometry.

Vector Representation in Geometry

Vectors are powerful tools in mathematics and geometry. They represent a direction and magnitude, crucial for tasks such as determining the distance and direction between two points. In the context of triangles, vectors help find and represent medians or other essential line segments in a triangle.

Consider the vector \( \vec{CM} \) from point \( C \) to midpoint \( M \) on side \( AB \). This vector is calculated as:

• \( \vec{CM} = (x_m - x_3, y_m - y_3) \)

Vectors not only represent simple paths between points but also can be scaled. For instance, to find a point two-thirds of the way along \( CM \), you use the operation:

• \( P = C + \frac{2}{3} \vec{CM} \)

This type of manipulation is crucial for determining points of balance, like centroids, as discussed in problems involving centroids of triangles.

Center of Mass in a Triangle

The center of mass, or centroid, of a triangle is where all of its medians intersect, creating a balance point within the shape. The centroid is mathematically significant because if a triangle were a physical object of uniform density, it would balance perfectly at its centroid.
The coordinates of the centroid \( G \) for triangle \( \Delta ABC \) are calculated using:

• \( G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \)

This calculation takes the average of the x-coordinates and y-coordinates of the vertices. The centroid is not only a point of symmetry; it is also vital in physics and engineering where balance and mass distribution are essential considerations. Understanding the center of mass allows for deeper insights into the mechanical and geometric properties of the triangle.