Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the length of the line segment joining the midpoints of the segments \(A B\) and \(C D\), where \(A=(1,3), B=(2,6)\), \(C=(4,7)\), and \(D=(3,4)\).

Step-by-Step Solution

Answer

The length of the segment between midpoints is \(\sqrt{5}\).

Step 1: Find the midpoint of segment AB

The midpoint formula for the segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). For segment AB, substitute \(A = (1, 3)\) and \(B = (2, 6)\) into the midpoint formula: \[ \left( \frac{1 + 2}{2}, \frac{3 + 6}{2} \right) = \left( \frac{3}{2}, \frac{9}{2} \right) \].So, the midpoint of AB is \(\left( \frac{3}{2}, \frac{9}{2} \right)\).

Step 2: Find the midpoint of segment CD

Again using the midpoint formula for the segment joining points \((x_1, y_1)\) and \((x_2, y_2)\), substitute \(C = (4, 7)\) and \(D = (3, 4)\): \[ \left( \frac{4 + 3}{2}, \frac{7 + 4}{2} \right) = \left( \frac{7}{2}, \frac{11}{2} \right) \].Thus, the midpoint of CD is \(\left( \frac{7}{2}, \frac{11}{2} \right)\).

Step 3: Calculate the distance between the midpoints

To find the distance between the midpoints \(M_1 = \left( \frac{3}{2}, \frac{9}{2} \right)\) and \(M_2 = \left( \frac{7}{2}, \frac{11}{2} \right)\), use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).Substitute the midpoint coordinates:\[d = \sqrt{\left( \frac{7}{2} - \frac{3}{2} \right)^2 + \left( \frac{11}{2} - \frac{9}{2} \right)^2}\]\[d = \sqrt{\left( 2 \right)^2 + \left( 1 \right)^2}\]\[d = \sqrt{4 + 1} = \sqrt{5}\].

Step 4: Conclude with the result

The length of the line segment joining the midpoints of segments AB and CD is \(\sqrt{5}\). This represents the Euclidean distance between the two calculated midpoints.

Key Concepts

Midpoint Formula

Understanding the midpoint formula is essential in coordinate geometry when determining a point that is exactly halfway between two other points on a plane. Imagine you have two endpoints of a line segment. The midpoint will be the point that divides this segment into two equal halves.
The formula is given by:

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

Where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two endpoints.Applying this, if you have segment AB with endpoints \(A = (1, 3)\) and \(B = (2, 6)\), you substitute these values into the formula:

• Midpoint of AB = \( \left( \frac{1 + 2}{2}, \frac{3 + 6}{2} \right) = \left( \frac{3}{2}, \frac{9}{2} \right) \)

This single point, \( \left( \frac{3}{2}, \frac{9}{2} \right) \), now perfectly represents the middle of segment AB. The same process is repeated with any other segment, such as CD here, to find its midpoint as well.

Distance Formula

The distance formula helps us find the length of the line segment connecting any two points in a plane. This is highly useful when measuring the exact separation between locations in geometry.
Mathematically, it is expressed as:

• Distance \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

To illustrate, when calculating the distance between the midpoints found earlier, \(M_1 = \left( \frac{3}{2}, \frac{9}{2} \right)\) and \(M_2 = \left( \frac{7}{2}, \frac{11}{2} \right)\), you simply replace \(x_1, y_1, x_2,\) and \(y_2\) with these coordinates in the formula:

• \(d = \sqrt{\left( \frac{7}{2} - \frac{3}{2} \right)^2 + \left( \frac{11}{2} - \frac{9}{2} \right)^2} \)
• Simplifying this, gives \(\sqrt{4 + 1} = \sqrt{5} \)

The outcome, \(\sqrt{5}\), is then the distance, or length, of the line segment between these midpoints.

Euclidean Distance

When working on geometry problems, one might encounter the term "Euclidean distance." Named after the ancient Greek mathematician Euclid, this concept captures the most straightforward way of measuring distance between two points, akin to using a ruler.
In essence, it is the "straight-line" distance, fully aligning with our intuitive understanding of physical space.
To compute Euclidean Distance is to employ exactly the distance formula mentioned above:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

In practice, recognize how each component under the square root calculates the differences in the x-coordinates and y-coordinates, squares them, and sums them, before taking the square root of the sum.
This method is crucial, as it precisely determines the shortest path in a flat, two-dimensional plane. Using Euclidean distance for the midpoints gives an accurate measurement of linear separation.
The result of \(\sqrt{5}\) between midpoints in the original exercise is a classic example of utilizing Euclidean principles in coordinate geometry.