Problem 47 - Precalculus Mathematics for Calculus

Question

(a) Sketch the parallelogram with vertices \(A(-2,-1)\) \(B(4,2), C(7,7),\) and \(D(1,4)\) (b) Find the midpoints of the diagonals of this parallelogram. (c) From part (b) show that the diagonals bisect each other.

Step-by-Step Solution

Answer

Both diagonals bisect each other at point \( \left( \frac{5}{2}, 3 \right) \).

Step 1: Plot the vertices

To sketch the parallelogram, start by plotting the given points on a coordinate plane. The points are \( A(-2,-1) \), \( B(4,2) \), \( C(7,7) \), and \( D(1,4) \). Connect the points in the respective order: \( A \to B \to C \to D \to A \) to form the parallelogram.

Step 2: Draw the diagonals

A diagonal is a line segment that connects two non-adjacent vertices of a polygon. Draw diagonal \( AC \) by connecting points \( A(-2,-1) \) and \( C(7,7) \). Draw diagonal \( BD \) by connecting points \( B(4,2) \) and \( D(1,4) \).

Step 3: Calculate the midpoint of diagonal AC

The midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \( \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \). For diagonal \( AC \), the endpoints are \((-2,-1)\) and \((7,7)\). Substitute these into the formula:\[\left( \frac{-2+7}{2}, \frac{-1+7}{2} \right) = \left( \frac{5}{2}, 3\right)\]

Step 4: Calculate the midpoint of diagonal BD

Using the same midpoint formula, calculate the midpoint of diagonal \( BD \) with endpoints \((4,2)\) and \((1,4)\):\[\left( \frac{4+1}{2}, \frac{2+4}{2} \right) = \left( \frac{5}{2}, 3\right)\]

Step 5: Conclude that the diagonals bisect each other

Both midpoints calculated are \( \left( \frac{5}{2}, 3 \right) \), confirming that the diagonals bisect each other at this point (the midpoints are the same). In a parallelogram, the diagonals always bisect each other, and this calculation verifies that property.

Key Concepts

Midpoint Formula

The Midpoint Formula is an essential tool in coordinate geometry, especially when dealing with line segments. When you want to find the center point of a line segment, the midpoint formula comes to the rescue. It's about averaging the coordinates of the endpoints of the line segment.

If you have a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \), you can find the midpoint by using the formula:

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

For instance, when we applied this to diagonal AC from the parallelogram, we averaged the x-coordinates \( -2 \) and \( 7 \) to get \( \frac{5}{2} \), and the y-coordinates \( -1 \) and \( 7 \) to get \( 3 \).

This point represents the exact middle of diagonal AC. The same process was used for diagonal BD, and naturally, the midpoints matched, illustrating a key property of parallelograms: their diagonals bisect each other.

Coordinate Geometry

Coordinate Geometry, also known as analytical geometry, is a branch of geometry where Cartesian coordinates are used to determine the positions of points. This system is extremely powerful as it allows geometric figures to be solved algebraically.

Every point in the plane is represented by an ordered pair \( (x, y) \), where \( x \) and \( y \) denote the distances of the point from the x-axis and y-axis, respectively. Let’s visualize:

• The line segment joining two points can be represented by its endpoints, like how diagonal AC is determined by the coordinates of points A \( (-2, -1) \) and C \( (7, 7) \).
• Operations such as determining the midpoint or distance between points utilize simple mathematical formulas within this coordinate system.

The beauty of coordinate geometry lies in how it seamlessly combines geometric and algebraic concepts, making it a tool for precise calculations and proofs such as proving that diagonals bisect each other in a parallelogram.

Diagonals of a Parallelogram

Diagonals are crucial components when studying parallelograms. In geometry, a diagonal refers to a line segment that connects two non-adjacent vertices in a polygon. In a parallelogram, there are always two distinct diagonals, each connecting a pair of opposite vertices.

In our earlier example, we named diagonals AC and BD:

• Diagonal AC connects vertex A with vertex C.
• Diagonal BD connects vertex B with vertex D.

Diagonals not only help determine congruence and symmetry in parallelograms, but they also explain inherent properties such as bisecting each other, which is unique to parallelograms. This means that a significant point on a parallelogram is the point where these two diagonals intersect, perfectly dividing each into two equal parts.

Bisecting Diagonals

When we say that the diagonals of a parallelogram bisect each other, it means that they intersect at a point that is equidistant from the endpoints of each diagonal. This is a defining characteristic of a parallelogram, and it is elegantly proven using coordinate geometry.

For example, let us take diagonal AC with midpoint \( \left( \frac{5}{2}, 3 \right) \) and diagonal BD with the same midpoint. This identical midpoint indicates that the diagonals each cut the other into two congruent segments.

This property can be generalized for any parallelogram:

• If the diagonals' midpoints are the same, they bisect each other.
• This property confirms the parallelogram's symmetry and is a key reason why midpoints are as important as vertices in understanding geometric families.

By using methods like the midpoint formula, students can easily verify this property of parallelograms in a coordinate plane, solidifying their understanding of geometry and algebra.