What is the distance in the standard \((x, y)\) coordinate plane between the points \((1,0)\) and \((0,5) ?\) A. 4 B. 6 C. 16 D. 36 E. \(\sqrt{26}\)

Understanding how to calculate the distance between two points is a fundamental skill in geometry and crucial for various standardized tests, like the ACT Math section. The distance formula, rooted in the Pythagorean theorem, is given by \(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This essentially calculates the length of the hypotenuse of a right triangle formed by connecting the two points and projecting perpendicular lines to the x-axis and y-axis from each point.
To apply the formula, simply identify the coordinates of the points involved, which we symbolize as \( (x_1,y_1) \) and \( (x_2,y_2) \). Distances are always positive in value, and the squaring and square roots ensure this in the distance formula. When you plug in the values for each point's x and y coordinates, you'll find the distance is a direct line from one point to another on the coordinate plane.

Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. In the context of geometry on a coordinate plane, we utilize concepts such as points, lines, distances, and angles to describe the location and relation of figures in space. The coordinate plane itself is composed of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
Each point in this plane is defined by an ordered pair \( (x, y) \), which represents its position relative to the two axes. Understanding how to maneuver through the coordinate plane, utilizing the distance formula, and interpreting the spatial relationships between points are key skills that have a multitude of applications in mathematics, physics, engineering, and beyond. Moreover, visualizing geometric problems helps to strengthen analytical thinking and problem-solving capabilities.

Effective preparation for the ACT Math section involves mastering a variety of mathematical concepts, including algebra, geometry, trigonometry, and probability. For geometry, particularly, knowing how to use the distance formula is essential since it allows test-takers to tackle questions about the lengths of segments on a coordinate plane swiftly.
During your ACT Math preparation, practice applying the distance formula in diverse problems to build speed and accuracy. Use timed quizzes to simulate test conditions and work on a mix of problems to familiarize yourself with the ACT's question format. Always remember to reduce the time spent on calculations by memorizing key formulas and theorems, including the distance formula and Pythagorean theorem, which are often the basis for a number of questions on the exam.

The Pythagorean theorem is a cornerstone of geometry, expressing a relationship in a right triangle between the lengths of the sides and the hypotenuse: \( a^2 + b^2 = c^2 \), where \(c\) is the hypotenuse, the side opposite the right angle, and \(a\) and \(b\) are the two other sides. This principle is not just theoretical; it's a practical tool used in various applications like navigation, construction, and analyzing polygons.
When solving for the distance between two points on a coordinate plane, we actually create a right triangle where the legs are parallel to the axes. The distance formula is an algebraic representation of the Pythagorean theorem, making it a direct application of this foundational theorem in a coordinate system. Recognizing this connection helps in understanding the geometric meaning behind the algebraic procedures used to find distances in the coordinate plane.