Problem 53 - Algebra and Trigonometry

Question

Use slopes to show that \(A(1,1), B(7,4), C(5,10),\) and \(D(-1,7)\) are vertices of a parallelogram.

Step-by-Step Solution

Answer

Slopes show both pairs of opposite sides are parallel, so it's a parallelogram.

Step 1: Calculate the Slope of Line AB

Slope of a line through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \(A(1, 1)\) and \(B(7, 4)\), the slope is \(m_{AB} = \frac{4 - 1}{7 - 1} = \frac{3}{6} = \frac{1}{2}\).

Step 2: Calculate the Slope of Line CD

For points \(C(5, 10)\) and \(D(-1, 7)\), the slope is \(m_{CD} = \frac{7 - 10}{-1 - 5} = \frac{-3}{-6} = \frac{1}{2}\).

Step 3: Verify Slope of Opposite Sides AB and CD

The slopes of line segments \(AB\) and \(CD\) are both \(\frac{1}{2}\), which means \(AB \parallel CD\).

Step 4: Calculate the Slope of Line BC

For points \(B(7, 4)\) and \(C(5, 10)\), the slope is \(m_{BC} = \frac{10 - 4}{5 - 7} = \frac{6}{-2} = -3\).

Step 5: Calculate the Slope of Line AD

For points \(A(1, 1)\) and \(D(-1, 7)\), the slope is \(m_{AD} = \frac{7 - 1}{-1 - 1} = \frac{6}{-2} = -3\).

Step 6: Verify Slope of Opposite Sides BC and AD

The slopes of line segments \(BC\) and \(AD\) are both \(-3\), which means \(BC \parallel AD\).

Step 7: Conclusion Based on Slope Calculations

Since both pairs of opposite sides \((AB \parallel CD)\) and \((BC \parallel AD)\) are parallel, the quadrilateral formed by points \(A, B, C,\) and \(D\) is a parallelogram.

Key Concepts

Parallelogram

A parallelogram is a four-sided figure, or quadrilateral, with opposite sides that are parallel to each other. This parallel alignment gives the shape some unique properties: opposite sides are equal in length and opposite angles are equal as well. Knowing these properties can help identify and calculate specific attributes of a parallelogram, making it an important shape in geometry. Imagine cutting a sheet of paper into a shape with two pairs of parallel lines. No matter how you change the angles of the lines, as long as both pairs stay parallel, the shape remains a parallelogram. To confirm whether a set of four points forms a parallelogram using coordinate geometry, we use the concept of slopes. The slopes, which measure steepness, must be the same for each pair of opposite sides. This means both pairs of opposite sides should have equal slopes, confirming they are parallel.

Coordinate Geometry

Coordinate geometry, or analytical geometry, combines algebra and geometry using a coordinate plane. It allows for easy calculations of distances and slopes between points, helping to determine shapes and their properties without necessarily measuring them directly.In a coordinate plane, each point is described by a pair of numbers known as coordinates. These numbers describe a point's unique position by indicating how far it is along the horizontal (x-axis) and the vertical (y-axis) lines.For example, point \((1, 1)\) stands at "1" along the x-axis and "1" along the y-axis. By connecting multiple points, we create shapes like lines or quadrilaterals. Using the coordinate geometry principles and formulas like the slope of a line, which is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\), you can study these shapes deeply and prove certain characteristics, such as whether a set of points forms a parallelogram.

Algebraic Calculation

Algebraic calculation involves using algebraic expressions and operations to solve problems or prove geometric propositions. This includes finding slopes, calculating distances, or verifying properties of geometric shapes like quadrilaterals.Slopes are a key aspect of algebraic calculation in geometry. The slope between two points is calculated using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This operation turns two coordinate points into a single number representing how steep a line is.In our example of a parallelogram, the algebraic calculation of slopes was crucial. We calculated the slopes of all four sides using the given points. The opposite sides had identical slopes, indicating they were parallel. This algebraic approach provides a precise and reliable method to confirm geometric properties, such as the parallel nature of sides in a parallelogram, without physically measuring them.