Problem 75 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the line through the midpoints of two sides of a triangle is parallel to the third side. Hint: You may assume that the triangle has vertices at \((0,0),(a, 0)\), and \((b, c)\).

Step-by-Step Solution

Answer

The line through the midpoints is parallel to the third side as both have a slope of zero.

Step 1: Identify Midpoints of Two Sides

Consider the triangle with vertices: \((0,0), (a,0), (b,c)\). First, find the midpoints of two sides of the triangle. We will find the midpoints of the sides connecting \((0,0)\) to \((b,c)\) and \((a,0)\) to \((b,c)\). The midpoint of the side connecting \((0,0)\) to \((b,c)\) is given by the formula:\[\left( \frac{0+b}{2}, \frac{0+c}{2} \right) = \left( \frac{b}{2}, \frac{c}{2} \right)\]The midpoint of the side connecting \((a,0)\) to \((b,c)\) is given by:\[\left( \frac{a+b}{2}, \frac{0+c}{2} \right) = \left( \frac{a+b}{2}, \frac{c}{2} \right)\]

Step 2: Find the Slope of Line Through Midpoints

Compute the slope of the line connecting the midpoints \(\left( \frac{b}{2}, \frac{c}{2} \right)\) and \(\left( \frac{a+b}{2}, \frac{c}{2} \right)\). The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substituting the midpoints, we have:\[m = \frac{\frac{c}{2} - \frac{c}{2}}{\frac{a+b}{2} - \frac{b}{2}} = \frac{0}{\frac{a}{2}} = 0\]

Step 3: Determine the Slope of the Third Side

Find the slope of the third side of the triangle, which is through points \((0,0)\) and \((a,0)\).Using the slope formula again:\[m = \frac{0 - 0}{a - 0} = 0\]Thus, the slope of the third side is also 0.

Step 4: Compare Slopes to Check Parallelism

The slope of the line through the midpoints is 0, and the slope of the third side \((0,0)\) to \((a,0)\) is also 0. Two lines that have the same slope are parallel to each other. Hence, the line through the midpoints is parallel to the third side of the triangle.

Key Concepts

Triangle Properties

Triangles are fundamental shapes in geometry, noted for their three edges and three vertices. Each type of triangle has unique properties which can be used to solve various problems.
Understanding the properties of triangles can help in understanding more complex concepts in geometry.

• A triangle's interior angles always add up to 180 degrees.
• The side opposite the largest angle is the longest.
• The sum of the lengths of any two sides must exceed the length of the third side.
• A right triangle contains one 90-degree angle, and its other two angles must add up to 90 degrees.

In coordinate geometry, triangles can be plotted using coordinates to further explore properties like midpoints and slopes, as shown in the exercise above.

Midpoint Formula

The midpoint formula is a helpful tool in coordinate geometry for finding the center point of a line segment.
This is especially useful in proofs or when trying to find specific geometric properties, like the one in the original exercise.
The formula is given by:\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]
Where

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the endpoints of the line segment.
• The resulting coordinates give the midpoint of the segment, dividing it into two equal parts.

Midpoints are useful for dividing triangles into smaller, manageable parts, and understanding symmetry within geometric shapes.

Slope of a Line

The slope of a line measures its steepness, indicating how much it inclines or declines horizontally.
It plays a crucial role in determining the parallelism or perpendicularity of lines in coordinate geometry.
The general formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Slope can take different forms, such as:

• Zero slope: indicates a horizontal line.
• Undefined slope: indicates a vertical line.
• Positive slope: line inclines upward.
• Negative slope: line declines downward.

In the exercise, the slopes of the lines being equal reveals that the lines are parallel.

• Lines with identical slopes do not intersect and are always equidistant.

Coordinate Geometry

Coordinate geometry, or analytical geometry, is where algebra meets geometry.
This branch of mathematics uses coordinates on a plane to solve geometric problems.
By using coordinate systems, we can employ algebraic techniques to study geometrical shapes and their properties.
In coordinate geometry:

• Points are defined using pairs of numbers, usually \( (x, y) \).
• Lines can be represented by linear equations.
• More complex shapes can be analyzed using quadratic equations or other polynomial equations.

By plotting triangles into a coordinate plane, we can employ useful principles like the midpoint formula and slope calculations to analyze properties.
This reveals deeper geometric insights and offers a visual representation to supplement understanding.