Right Triangle Ordered Pair: Find The Missing Coordinate

Let's dive into the world of coordinate geometry and explore how to find the missing ordered pair that completes a right triangle. This problem involves understanding the properties of right triangles, particularly the Pythagorean theorem and the concept of perpendicular lines. We will break down the problem step-by-step, making it easy to follow and understand. So, let's get started!

Understanding the Problem: Right Triangles and Coordinate Geometry

In coordinate geometry, we often deal with shapes and figures plotted on a coordinate plane. A right triangle, as you might already know, is a triangle with one angle measuring 90 degrees. This right angle is crucial because it allows us to use the Pythagorean theorem, a fundamental concept in geometry. The Pythagorean theorem 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 (legs). Mathematically, this is expressed as: $a^2 + b^2 = c^2$, where c is the length of the hypotenuse, and a and b are the lengths of the legs.

Now, let's consider how this applies to coordinate geometry. When we have points on a coordinate plane, we can calculate the distance between any two points using the distance formula. This formula is derived from the Pythagorean theorem itself! If we have two points, $(x_1, y_1)$ and $(x_2, y_2)$, the distance, d, between them is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This formula essentially calculates the length of the line segment connecting the two points. In the context of our problem, we're given two points, (-5, 4) and (-2, 1), and we need to find a third point that, when connected to the other two, forms a right triangle. To do this, we'll need to consider the distances between these points and how they relate to the Pythagorean theorem. Furthermore, the concept of perpendicular lines comes into play. Two lines are perpendicular if they intersect at a right angle. In coordinate geometry, the slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other. This means that if one line has a slope of m, a line perpendicular to it will have a slope of -1/m. Understanding this relationship will help us determine if the lines formed by our points create a right angle.

Step-by-Step Solution: Finding the Right Ordered Pair

Let's break down the process of finding the ordered pair that forms a right triangle with the given points (-5, 4) and (-2, 1). We'll use a combination of the distance formula and the concept of perpendicular lines to arrive at the solution. Here's a step-by-step approach:

1. Calculate the Distance Between the Given Points:

First, we need to find the distance between the two points we already have: (-5, 4) and (-2, 1). We'll use the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Plugging in the coordinates, we get:

$d = \sqrt{(-2 - (-5))^2 + (1 - 4)^2} = \sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$

So, the distance between (-5, 4) and (-2, 1) is $3\sqrt{2}$ units. This will be one side of our potential right triangle.

2. Test Each Ordered Pair:

Now, we need to test each of the given options (A, B, C, and D) to see which one, when combined with the original two points, forms a right triangle. We'll do this by calculating the distances between the new point and the original two points. Then, we'll check if the Pythagorean theorem holds true for these three distances. Let's go through each option:

• Option A: (1, -5)

  • Distance between (-5, 4) and (1, -5):

  $d_1 = \sqrt{(1 - (-5))^2 + (-5 - 4)^2} = \sqrt{(6)^2 + (-9)^2} = \sqrt{36 + 81} = \sqrt{117}$

  • Distance between (-2, 1) and (1, -5):

  $d_2 = \sqrt{(1 - (-2))^2 + (-5 - 1)^2} = \sqrt{(3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45}$

  • Check Pythagorean theorem: $(3\sqrt{2})^2 + (\sqrt{45})^2 = (\sqrt{117})^2$?

  $18 + 45 = 117$? No, 63 ≠ 117. So, option A is not the correct answer.

• Option B: (-5, -2)

  • Distance between (-5, 4) and (-5, -2):

  $d_1 = \sqrt{(-5 - (-5))^2 + (-2 - 4)^2} = \sqrt{(0)^2 + (-6)^2} = \sqrt{36} = 6$

  • Distance between (-2, 1) and (-5, -2):

  $d_2 = \sqrt{(-5 - (-2))^2 + (-2 - 1)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$

  • Check Pythagorean theorem: $(3\sqrt{2})^2 + (3\sqrt{2})^2 = (6)^2$?

  $18 + 18 = 36$? Yes, 36 = 36. So, option B could be the correct answer. We still need to verify it using the slope method to be absolutely sure.

• Option C: (-5, 1)

  • Distance between (-5, 4) and (-5, 1):

  $d_1 = \sqrt{(-5 - (-5))^2 + (1 - 4)^2} = \sqrt{(0)^2 + (-3)^2} = \sqrt{9} = 3$

  • Distance between (-2, 1) and (-5, 1):

  $d_2 = \sqrt{(-5 - (-2))^2 + (1 - 1)^2} = \sqrt{(-3)^2 + (0)^2} = \sqrt{9} = 3$

  • Check Pythagorean theorem: $(3\sqrt{2})^2 + (3)^2 = (3)^2$?

  $18 + 9 = 9$? No, 27 ≠ 9. So, option C is not the correct answer.

• Option D: (4, -2)

  • Distance between (-5, 4) and (4, -2):

  $d_1 = \sqrt{(4 - (-5))^2 + (-2 - 4)^2} = \sqrt{(9)^2 + (-6)^2} = \sqrt{81 + 36} = \sqrt{117}$

  • Distance between (-2, 1) and (4, -2):

  $d_2 = \sqrt{(4 - (-2))^2 + (-2 - 1)^2} = \sqrt{(6)^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45}$

  • Check Pythagorean theorem: $(3\sqrt{2})^2 + (\sqrt{45})^2 = (\sqrt{117})^2$?

  $18 + 45 = 117$? No, 63 ≠ 117. So, option D is not the correct answer.

3. Verify with Slopes (for Option B):

Since option B satisfied the Pythagorean theorem, we need to confirm that it indeed forms a right triangle by checking if any two sides are perpendicular. We'll do this by calculating the slopes of the lines formed by the points and checking if any two slopes are negative reciprocals of each other.

• Slope between (-5, 4) and (-2, 1):

  $m_1 = (1 - 4) / (-2 - (-5)) = -3 / 3 = -1$

• Slope between (-5, 4) and (-5, -2):

  $m_2 = (-2 - 4) / (-5 - (-5)) = -6 / 0$ This slope is undefined, which means the line is vertical.

• Slope between (-2, 1) and (-5, -2):

  $m_3 = (-2 - 1) / (-5 - (-2)) = -3 / -3 = 1$

Since $m_1$ (-1) and $m_3$ (1) are negative reciprocals of each other, the lines formed by these points are perpendicular, and we have a right triangle. Also, a vertical line (undefined slope) is perpendicular to a horizontal line (slope of 0). In our case, the line with undefined slope is perpendicular to the line segment with slope 1.

4. Conclusion:

After thoroughly analyzing each option, we have determined that the ordered pair (-5, -2) (Option B) is the correct answer. It satisfies both the Pythagorean theorem and the condition of perpendicular lines, confirming that it forms a right triangle with the points (-5, 4) and (-2, 1).

Alternative Method: Using Slopes Directly

While the distance formula and Pythagorean theorem method is effective, there's another approach that directly utilizes the concept of slopes to identify right triangles. This method can sometimes be faster, especially if you're comfortable working with slopes.

1. Calculate the Slope Between the Given Points:

As before, let's start by finding the slope of the line segment formed by the points (-5, 4) and (-2, 1):

$m_1 = (1 - 4) / (-2 - (-5)) = -3 / 3 = -1$

2. Calculate Slopes for Each Option:

Now, for each option, we'll calculate the slopes of the lines formed by connecting that point to the original two points. We're looking for a pair of slopes where one is the negative reciprocal of the other.

• Option A: (1, -5)

  • Slope between (-5, 4) and (1, -5):

  $m_{A1} = (-5 - 4) / (1 - (-5)) = -9 / 6 = -3/2$

  • Slope between (-2, 1) and (1, -5):

  $m_{A2} = (-5 - 1) / (1 - (-2)) = -6 / 3 = -2$

  Are -3/2 and -2 negative reciprocals? No.

• Option B: (-5, -2)

  • Slope between (-5, 4) and (-5, -2):

  $m_{B1} = (-2 - 4) / (-5 - (-5)) = -6 / 0$ (undefined - vertical line)

  • Slope between (-2, 1) and (-5, -2):

  $m_{B2} = (-2 - 1) / (-5 - (-2)) = -3 / -3 = 1$

  A vertical line is perpendicular to a line with slope 0. The slope between (-5,4) and (-2,1) is -1. The negative reciprocal of -1 is 1, which is the slope we found ($m_{B2}$). So, this is a potential right triangle.

• Option C: (-5, 1)

  • Slope between (-5, 4) and (-5, 1):

  $m_{C1} = (1 - 4) / (-5 - (-5)) = -3 / 0$ (undefined - vertical line)

  • Slope between (-2, 1) and (-5, 1):

  $m_{C2} = (1 - 1) / (-5 - (-2)) = 0 / -3 = 0$ (horizontal line)

  A vertical and horizontal line are perpendicular, but we need to check if the hypotenuse is formed correctly. The slope between the original points is -1, and neither of these lines has a slope that is a negative reciprocal of -1.

• Option D: (4, -2)

  • Slope between (-5, 4) and (4, -2):

  $m_{D1} = (-2 - 4) / (4 - (-5)) = -6 / 9 = -2/3$

  • Slope between (-2, 1) and (4, -2):

  $m_{D2} = (-2 - 1) / (4 - (-2)) = -3 / 6 = -1/2$

  Are -2/3 and -1/2 negative reciprocals? No.

3. Conclusion:

Using the slope method, we again find that Option B, (-5, -2), is the ordered pair that forms a right triangle with the given points. The slopes between the points confirm that two sides are perpendicular.

Key Takeaways and Practice Problems

• Understanding the Pythagorean Theorem: The Pythagorean theorem is a cornerstone of geometry and is crucial for solving problems involving right triangles. Remember that $a^2 + b^2 = c^2$, where c is the hypotenuse.
• Distance Formula: The distance formula, $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, allows you to calculate the distance between any two points on a coordinate plane. This is essential for applying the Pythagorean theorem in coordinate geometry.
• Slopes of Perpendicular Lines: Two lines are perpendicular if their slopes are negative reciprocals of each other. If one line has a slope of m, a line perpendicular to it will have a slope of -1/m. This concept is vital for identifying right angles in coordinate geometry.
• Multiple Solution Methods: We explored two methods for solving this type of problem: the distance formula with the Pythagorean theorem and the slope method. Being familiar with both approaches can provide you with flexibility and help you choose the most efficient method for a given problem.

By mastering these concepts and practicing regularly, you'll become proficient in solving coordinate geometry problems involving right triangles. Remember to visualize the problem, break it down into manageable steps, and double-check your calculations. Coordinate geometry is a fascinating branch of mathematics that combines geometric principles with algebraic techniques. Embrace the challenge, and you'll find it both rewarding and intellectually stimulating.

For further learning and practice, you can explore resources like Khan Academy's Geometry Section, which offers comprehensive lessons and practice exercises on various geometric topics.