Find The Third Vertex Coordinates Of An Equilateral Triangle If Two Vertices Are (2, 4) And (2, 6).

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Finding the third vertex of an equilateral triangle when given two vertices is a classic problem in coordinate geometry. This problem combines geometric properties with algebraic techniques, offering a fascinating challenge for students and enthusiasts alike. This comprehensive guide will walk you through the steps to solve this problem, explore the underlying concepts, and provide a detailed explanation to ensure a thorough understanding. Understanding equilateral triangles is crucial, as they form the basis for many geometric constructions and have significant applications in various fields, including engineering, architecture, and computer graphics. Before diving into the solution, let’s first recap the key properties of equilateral triangles and the distance formula, which are essential tools for solving this problem.

Understanding Equilateral Triangles

An equilateral triangle is a triangle in which all three sides are of equal length and all three angles are equal, each measuring 60 degrees. This symmetry leads to several important properties that are useful in geometric constructions and calculations. In this section, we will delve into the key characteristics of equilateral triangles, focusing on their sides, angles, symmetry, and the relationships between them. These properties not only help in solving geometric problems but also provide a deeper understanding of geometric principles. The equal sides and angles of an equilateral triangle create a harmonious balance, making it a fundamental shape in geometry. Knowing these properties allows us to use the distance formula effectively and apply geometric theorems to find unknown vertices. By mastering these concepts, you’ll be well-equipped to tackle complex problems involving equilateral triangles and coordinate geometry. Additionally, we’ll explore the symmetry inherent in these triangles, which plays a crucial role in visualizing and solving related problems. The symmetry of equilateral triangles means they can be divided into congruent parts, which simplifies many geometric proofs and constructions. This symmetry is not just a visual characteristic but also a mathematical property that can be exploited to solve problems more efficiently. In essence, equilateral triangles are more than just shapes; they are geometric entities with unique properties that make them invaluable in both theoretical mathematics and practical applications. Recognizing and understanding these properties is the first step in mastering geometry and its applications.

Key Properties of Equilateral Triangles

  1. Equal Sides: All three sides of an equilateral triangle are of the same length. This is the defining characteristic of an equilateral triangle and is fundamental to many geometric proofs and constructions.
  2. Equal Angles: Each angle in an equilateral triangle measures 60 degrees. The sum of the angles in any triangle is 180 degrees, and since all angles are equal in an equilateral triangle, each angle must be 180/3 = 60 degrees.
  3. Symmetry: Equilateral triangles possess a high degree of symmetry. They have three lines of symmetry, each passing through a vertex and the midpoint of the opposite side. This symmetry simplifies many geometric problems.
  4. Altitude and Median: The altitude (height) and median (line segment from a vertex to the midpoint of the opposite side) coincide in an equilateral triangle. This property is useful for calculations involving area and height.
  5. Center: The centroid, circumcenter, incenter, and orthocenter of an equilateral triangle are all the same point. This unique characteristic simplifies many calculations and constructions.

Distance Formula

The distance d between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane is given by the distance formula:

d=(x2−x1)2+(y2−y1)2\qquad d = \sqrt{(x₂ - x₁)² + (y₂ - y₁)²}

This formula is derived from the Pythagorean theorem and is essential for calculating the lengths of the sides of the triangle. It allows us to translate geometric distances into algebraic expressions, making it a crucial tool for solving problems in coordinate geometry. The distance formula is not only applicable to triangles but is a fundamental concept in various areas of mathematics, including calculus and linear algebra. Understanding and applying the distance formula correctly is essential for accurately determining distances and solving geometric problems. It provides a bridge between algebra and geometry, allowing us to analyze geometric shapes using algebraic methods. This formula is especially useful when dealing with coordinates and is a cornerstone of analytic geometry.

Problem Statement

Given two vertices of an equilateral triangle, A(2, 4) and B(2, 6), find the coordinates of the third vertex, C(x, y). This problem requires us to use the properties of equilateral triangles, the distance formula, and some algebraic manipulation to determine the unknown coordinates. Understanding the problem statement is the first step toward finding a solution. We know that all sides of an equilateral triangle are equal, so the distance between any two vertices must be the same. This fact will be crucial in setting up equations and solving for the unknown coordinates. The problem also highlights the interplay between geometry and algebra, as we will be using geometric properties to create algebraic equations. Furthermore, this type of problem often has multiple solutions, reflecting the symmetry of the equilateral triangle. Recognizing this possibility is important for ensuring a comprehensive solution.

Solution Steps

To solve this problem, we will follow these steps:

  1. Calculate the distance between the given vertices A and B. This distance will be the length of each side of the equilateral triangle.
  2. Set up equations using the distance formula to represent the distances AC and BC in terms of the unknown coordinates (x, y) of vertex C.
  3. Equate the distances AB, AC, and BC, as they are all equal in an equilateral triangle.
  4. Solve the resulting system of equations to find the possible coordinates of vertex C.

Step 1: Calculate the Distance Between Vertices A and B

Using the distance formula, the distance d between A(2, 4) and B(2, 6) is:

d=(2−2)2+(6−4)2=0+22=4=2\qquad d = \sqrt{(2 - 2)² + (6 - 4)²} = \sqrt{0 + 2²} = \sqrt{4} = 2

So, the length of each side of the equilateral triangle is 2 units. This initial calculation sets the scale for the rest of the problem. Knowing the side length allows us to use this value as a reference when setting up and solving equations for the coordinates of the third vertex. This step is crucial because it converts the geometric condition of equal sides into a quantifiable value that can be used in algebraic equations. The distance calculated here will be the basis for comparison when we calculate the distances AC and BC. Furthermore, this calculation highlights the importance of the distance formula in solving coordinate geometry problems. It demonstrates how a geometric concept (distance) can be represented and calculated using algebraic tools.

Step 2: Set Up Equations for Distances AC and BC

Let C(x, y) be the third vertex. The distance between A(2, 4) and C(x, y) is:

AC=(x−2)2+(y−4)2\qquad AC = \sqrt{(x - 2)² + (y - 4)²}

Similarly, the distance between B(2, 6) and C(x, y) is:

BC=(x−2)2+(y−6)2\qquad BC = \sqrt{(x - 2)² + (y - 6)²}

These equations represent the distances AC and BC in terms of the unknown coordinates x and y. Setting up these equations is a critical step in translating the geometric problem into an algebraic one. The distance formula is applied here to express the side lengths in terms of the coordinates, allowing us to form equations that can be solved algebraically. These equations encapsulate the condition that the sides of the triangle must have a specific length (which we found in Step 1). By using the distance formula, we transform the geometric property of equal side lengths into algebraic equations, setting the stage for solving for the unknown coordinates of vertex C. This step demonstrates the power of coordinate geometry, which allows us to use algebraic methods to solve geometric problems. The expressions for AC and BC will be used in subsequent steps to form a system of equations that can be solved for x and y.

Step 3: Equate the Distances AB, AC, and BC

Since the triangle is equilateral, AB = AC = BC. We know AB = 2, so we have:

AC=(x−2)2+(y−4)2=2\qquad AC = \sqrt{(x - 2)² + (y - 4)²} = 2

BC=(x−2)2+(y−6)2=2\qquad BC = \sqrt{(x - 2)² + (y - 6)²} = 2

Equating the distances is a fundamental step in solving for the coordinates of the third vertex. This step utilizes the defining property of an equilateral triangle: all sides are equal. By setting the distances AC and BC equal to the known side length AB, we create a system of equations that can be solved to find the unknown coordinates of vertex C. This approach transforms the geometric condition of equal sides into algebraic equations, allowing us to use algebraic techniques to find the solution. The equations formed in this step are crucial because they capture the geometric constraints of the problem in an algebraic form. Solving these equations will give us the possible locations of the third vertex, which satisfies the condition of forming an equilateral triangle with the given vertices A and B. This step also highlights the importance of understanding the properties of geometric shapes and how these properties can be used to solve problems.

Step 4: Solve the System of Equations

Squaring both equations, we get:

(x−2)2+(y−4)2=4\qquad (x - 2)² + (y - 4)² = 4 (Equation 1)

(x−2)2+(y−6)2=4\qquad (x - 2)² + (y - 6)² = 4 (Equation 2)

Subtracting Equation 2 from Equation 1, we get:

(y−4)2−(y−6)2=0\qquad (y - 4)² - (y - 6)² = 0

(y2−8y+16)−(y2−12y+36)=0\qquad (y² - 8y + 16) - (y² - 12y + 36) = 0

4y−20=0\qquad 4y - 20 = 0

y=5\qquad y = 5

Substituting y = 5 into Equation 1:

(x−2)2+(5−4)2=4\qquad (x - 2)² + (5 - 4)² = 4

(x−2)2+1=4\qquad (x - 2)² + 1 = 4

(x−2)2=3\qquad (x - 2)² = 3

x−2=±3\qquad x - 2 = ±\sqrt{3}

x=2±3\qquad x = 2 ± \sqrt{3}

Thus, the possible coordinates for vertex C are (2+3,5)(2 + \sqrt{3}, 5) and (2−3,5)(2 - \sqrt{3}, 5). Solving the system of equations is the core algebraic part of the problem. This step involves manipulating the equations derived from the distance formula to isolate the unknown variables x and y. By squaring the equations, subtracting them, and simplifying, we find the possible values for y. Substituting these values back into one of the original equations allows us to solve for x. This process demonstrates the power of algebraic manipulation in solving geometric problems. The solutions obtained represent the possible locations of the third vertex that would form an equilateral triangle with the given vertices. This step requires a solid understanding of algebraic techniques, such as expanding squares, simplifying equations, and solving quadratic equations. The solutions obtained are not just numerical answers but also geometric points that satisfy the given conditions.

Possible Solutions

The two possible coordinates for the third vertex C are:

  1. (2+3,5)(2 + \sqrt{3}, 5)
  2. (2−3,5)(2 - \sqrt{3}, 5)

These two solutions reflect the symmetry of the equilateral triangle. There are two possible positions for the third vertex that will form an equilateral triangle with the given vertices A and B. This result is a direct consequence of the geometric properties of equilateral triangles. The two solutions are symmetrically positioned with respect to the line segment AB. The existence of two solutions highlights the importance of considering all possibilities when solving geometric problems. Each solution represents a valid vertex that, when connected to the given vertices A and B, forms an equilateral triangle. This step provides a clear and concise answer to the problem, giving the coordinates of the possible locations for the third vertex. The two solutions are equally valid and demonstrate the complete set of possible answers. This outcome is not uncommon in geometric problems, and understanding the symmetry of the shape can often help in predicting the number of solutions.

Conclusion

Finding the third vertex of an equilateral triangle given two vertices is a problem that beautifully combines geometry and algebra. By using the distance formula and the properties of equilateral triangles, we can set up and solve a system of equations to find the possible coordinates of the third vertex. This problem not only reinforces our understanding of these concepts but also illustrates the power of analytical methods in solving geometric problems. The steps involved in solving this problem provide a comprehensive approach to tackling similar geometric challenges. From understanding the properties of shapes to applying the distance formula and solving algebraic equations, this process demonstrates the interconnectedness of mathematical concepts. The two possible solutions highlight the symmetry inherent in geometric shapes and the importance of considering all possibilities. This problem serves as a valuable exercise in developing problem-solving skills and deepening the understanding of geometry and algebra. By mastering these techniques, students can approach more complex geometric problems with confidence and precision.

This guide provides a detailed and comprehensive explanation of how to find the third vertex of an equilateral triangle given two vertices. By understanding the properties of equilateral triangles and applying the distance formula, you can confidently solve this type of problem. Remember to always consider all possible solutions and to check your work to ensure accuracy.