Question

What is the distance, in coordinate units, between the points \((-3,5)\) and \((4,-1)\) in the standard \((x, y)\) coordinate plane? F. \(\sqrt{13}\) G. \(\sqrt{17}\) H. \(\sqrt{85}\) J. 13 K. 85

Short answer

Answer: The distance between the points (-3,5) and (4,-1) in the coordinate plane is √85.

Step-by-step solution

Identify the coordinates of both points

We are given the coordinates of the two points: Point A: \((-3,5)\) - \(x_1 = -3\) - \(y_1 = 5\) Point B: \((4,-1)\) - \(x_2 = 4\) - \(y_2 = -1\)

Use the distance formula

Insert the coordinates from step 1 into the formula: $$ Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ = \sqrt{(4 - (-3))^2 + (-1 - 5)^2} $$

Calculate the distance

Simplify the expression inside the square root: $$ Distance = \sqrt{(4 + 3)^2 + (-1 - 5)^2} \\ = \sqrt{7^2 + (-6)^2} \\ = \sqrt{49 + 36} \\ = \sqrt{85} $$ So, the distance between the points \((-3,5)\) and \((4,-1)\) in the coordinate plane is \(\sqrt{85}\). The correct answer is H. \(\sqrt{85}\).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This method provides a connection between algebra and geometry through graphs of lines and curves. It enables us to utilize algebraic equations to describe geometric figures, thereby offering a powerful tool to understand the properties and relationships of shapes.

In the context of the distance formula, coordinate geometry allows us to calculate the distance between any two points on a plane. A coordinate plane is defined with two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is identified by an ordered pair of numbers (x, y), known as coordinates. The distance formula, derived from the Pythagorean theorem, is applied when we want to know the straight line distance between two such points.

The distance formula is:
\[\begin{equation}Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\end{equation}\]
where \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two points. This formula is essential in coordinate geometry because it provides a uniform method to measure distance regardless of the points' locations on the plane.

ACT Math Preparation

Preparing for the ACT Math test requires a solid understanding of many fundamental concepts in mathematics, one of which is coordinate geometry. The ACT will often include questions that involve the distance formula, slope-intercept form, midpoints, and other principles related to the coordinate plane. To perform well on these questions, it is crucial to have a systematic approach to problem-solving and a clear comprehension of coordinate geometry principles.

Strategies for ACT Math preparation include:

• Mastering the basic formulas and theorems, like the distance formula and the Pythagorean theorem.
• Practicing with multiple sets of ACT-like questions to become familiar with the format and types of problems you will encounter.
• Understanding how to quickly plot points and interpret the coordinate plane, as graphing is a common requirement.
• Developing problem-solving skills to approach questions logically and efficiently.

Incorporating the distance formula into your ACT Math study routines can significantly enhance your proficiency and confidence in tackling coordinate geometry questions on test day.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states that in a right 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. This theorem can be expressed as:
\[\begin{equation}a^2 + b^2 = c^2\end{equation}\]
where \(a\) and \(b\) are the lengths of the legs of the triangle, and \(c\) denotes the length of the hypotenuse.

When applied to coordinate geometry and the distance formula, the Pythagorean theorem helps us to understand why the formula works. If we consider the horizontal and vertical distances between two points as the legs of a right triangle, then the distance between the points is the hypotenuse. Thus, finding the distance between two points on a coordinate plane is an application of the Pythagorean theorem, where the sides of the triangle are the differences in the x and y coordinates, and the hypotenuse represents the distance we are trying to find.