Problem 116 - Algebra and Trigonometry

Question

Exercises \(116-118\) will help you prepare for the material covered in the next section. Use the distance formula to determine if the line segment with endpoints \((-3,-3)\) and \((0,3)\) has the same length as the line segment with endpoints \((0,0)\) and \((3,6)\)

Step-by-Step Solution

Answer

By plugging in the coordinates into the distance formula, both lengths are found to be \( \sqrt{36} \) or \( 6 \). Therefore, yes, these line segments have the same length.

Step 1: Calculate length of first line segment

Use the distance formula to calculate the length of the first line segment. For the line segment with endpoints \((-3,-3)\) and \((0,3)\), the formula becomes \(\sqrt{(0-(-3))^2 + (3-(-3))^2}\)

Step 2: Calculate length of second line segment

Similarly, use the distance formula to calculate the length of the second line segment. For the line segment with endpoints \((0,0)\) and \((3,6)\), the formula becomes \(\sqrt{(3-0)^2 + (6-0)^2}\)

Step 3: Compare lengths of the two line segments

Finally, compare the two calculated lengths. If they are equal, then the line segments have the same length, otherwise they do not.

Key Concepts

Line Segment Length

The line segment length measures the distance between two points in a Euclidean space. In geometry, this concept is fundamental for understanding the structure of shapes and for solving various problems related to size and position. The length of a line segment can be visualized as the straight path that connects its endpoints without any bends or curves.

When it comes to finding this length on a coordinate plane, we employ the distance formula, which is directly derived from the Pythagorean theorem. This formula is expressed as: \[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \(x_1, y_1)\) and \(x_2, y_2)\) are the coordinates of the endpoints of the line segment. Calculating the length of a line segment is an essential skill, especially when handling problems where lengths of various segments are compared, or when assessing the congruency of two segments.

Coordinate Geometry

In coordinate geometry, also known as analytic geometry, we study geometrical shapes and figures using a system of coordinates. The basic idea is to translate geometric problems into algebraic ones by placing them in a coordinate plane. This plane is two-dimensional and consists of a horizontal axis (x-axis) and a vertical axis (y-axis), which intersect at the origin (0,0).

Each point in this plane can be specified by an ordered pair of numbers, (x, y), that represents its coordinates. Using these coordinates, we can calculate the distance between points, the slopes of lines, and the equations of various geometric shapes. Coordinate geometry is indispensable for analyzing spatial relationships and solving complex problems with precision, which would be much harder to tackle with pure Euclidean geometry alone.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry, especially when dealing with right triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as: \[ a^2 + b^2 = c^2 \]

Application in Distance Calculation

By treating the line segment as the hypotenuse of a right triangle and the differences in the x and y coordinates as the other two sides, we can apply the Pythagorean theorem directly to calculate the segment's length. This approach is not only used to solve problems related to triangles but also to determine the distance between any two points in a plane. The theorem provides a simple yet profound connection between algebra and geometry that is essential for understanding and solving numerous geometric problems.