Right Triangle Vertices: Prove (14,10), (11,13), & (2,-2)

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Proving Points Form a Right-Angled Triangle: A Comprehensive Guide

Hey guys! Let's dive into a fun and important concept in coordinate geometry: proving that certain points are the vertices of a right-angled triangle. We'll take a specific example – the points (14, 10), (11, 13), and (2, -2) – and walk through the steps to demonstrate that they indeed form a right-angled triangle. This isn't just a math exercise; it's about understanding fundamental geometric principles and how they translate into algebraic calculations. So, buckle up, and let’s get started!

Understanding the Basics: Right-Angled Triangles and Coordinate Geometry

Before we jump into the proof, it’s crucial to have a solid grasp of what a right-angled triangle is and how we can use coordinate geometry to analyze geometric shapes. A right-angled triangle, as you probably know, is a triangle that has one angle measuring exactly 90 degrees. This right angle is the key characteristic that we’ll be looking for in our points.

In coordinate geometry, we use the Cartesian plane (the x-y plane) to represent points and shapes. Each point is defined by its coordinates (x, y). To prove that points form a right-angled triangle, we often rely on two main concepts:

  1. The Distance Formula: This formula helps us calculate the distance between two points in the coordinate plane. If we have two points, (x1, y1) and (x2, y2), the distance ‘d’ between them is given by:

    d = √((x2 - x1)² + (y2 - y1)²)

    This formula is derived from the Pythagorean theorem, which is actually the second key concept we'll use.

  2. The Pythagorean Theorem: This theorem 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. If ‘a’ and ‘b’ are the lengths of the two shorter sides (legs) and ‘c’ is the length of the hypotenuse, then:

    a² + b² = c²

    The Pythagorean Theorem is extremely important. We'll calculate the distances between our points using the distance formula and then check if these distances satisfy the Pythagorean theorem. If they do, we've proven that the points form a right-angled triangle!

Step-by-Step Proof: Showing (14, 10), (11, 13), and (2, -2) Form a Right Triangle

Okay, let’s get to the fun part! We'll systematically prove that the points (14, 10), (11, 13), and (2, -2) are the vertices of a right-angled triangle. To make things clear, let’s label these points:

  • A = (14, 10)
  • B = (11, 13)
  • C = (2, -2)

Our strategy is straightforward: we’ll calculate the distances between each pair of points (AB, BC, and CA) using the distance formula. Then, we’ll check if the squares of these distances satisfy the Pythagorean theorem.

1. Calculate the Distance Between Points A and B (AB)

Using the distance formula:

AB = √((11 - 14)² + (13 - 10)²)

AB = √((-3)² + (3)²)

AB = √(9 + 9)

AB = √18

So, the distance between points A and B is √18 units.

2. Calculate the Distance Between Points B and C (BC)

Again, using the distance formula:

BC = √((2 - 11)² + (-2 - 13)²)

BC = √((-9)² + (-15)²)

BC = √(81 + 225)

BC = √306

Therefore, the distance between points B and C is √306 units.

3. Calculate the Distance Between Points C and A (CA)

One more time with the distance formula:

CA = √((14 - 2)² + (10 - (-2))²)

CA = √((12)² + (12)²)

CA = √(144 + 144)

CA = √288

Hence, the distance between points C and A is √288 units.

4. Check if the Pythagorean Theorem Holds

Now comes the crucial part: Let's see if the squares of these distances satisfy the Pythagorean theorem. We have:

  • AB = √18, so AB² = 18
  • BC = √306, so BC² = 306
  • CA = √288, so CA² = 288

We need to check if a² + b² = c², where ‘c’ is the longest side (the potential hypotenuse). In this case, BC is the longest side since √306 is the largest distance. Let’s see if AB² + CA² = BC²:

18 + 288 = 306

Wow, it works! Since AB² + CA² = BC², the Pythagorean theorem holds true for these points. This strongly indicates that triangle ABC is a right-angled triangle.

Conclusion: Q.E.D. (Quod Erat Demonstrandum)

We’ve done it! By systematically applying the distance formula and the Pythagorean theorem, we have successfully demonstrated that the points (14, 10), (11, 13), and (2, -2) are indeed the vertices of a right-angled triangle. This process highlights the powerful connection between algebra and geometry. We used algebraic formulas to calculate distances and then applied a fundamental geometric theorem to reach our conclusion.

Key Takeaways:

  • The distance formula is essential for finding the distance between two points in the coordinate plane.
  • The Pythagorean theorem is the cornerstone for proving right-angled triangles.
  • Coordinate geometry allows us to analyze geometric shapes using algebraic methods.

So, the next time you encounter a similar problem, remember the steps we followed: calculate the distances, check the Pythagorean theorem, and you’ll be well on your way to proving geometric properties! Keep practicing, and you’ll become a pro at coordinate geometry in no time!