Solving Angle And Right Triangle Problems In Coordinate Geometry

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This comprehensive guide delves into two essential concepts in coordinate geometry: determining the slope of a line given the angle between it and another line, and verifying whether a triangle formed by three given points is a right triangle. We will explore the underlying principles, provide step-by-step solutions, and illustrate these concepts with examples. Understanding these concepts is crucial for various applications in mathematics, physics, engineering, and computer graphics.

Determining the Slope of a Line Given the Angle Between Two Lines

The slope of a line is a fundamental concept that describes its steepness and direction. It is defined as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. The slope is often denoted by the letter 'm'. The equation of a line in slope-intercept form is given by y = mx + c, where 'm' is the slope and 'c' is the y-intercept (the point where the line intersects the y-axis).

When two lines intersect, they form an angle between them. The angle between two lines is related to their slopes. If we know the angle between two lines and the slope of one line, we can determine the slope of the other line. This relationship is derived from the tangent of the angle between the lines.

The Formula

Let m1 and m2 be the slopes of two lines, and let θ be the angle between them. The formula relating the slopes and the angle is:

tan θ = |(m1 - m2) / (1 + m1 * m2)|

Where:

  • tan θ is the tangent of the angle θ.
  • m1 is the slope of the first line.
  • m2 is the slope of the second line.
  • The absolute value ensures that we consider the acute angle between the lines.

Step-by-Step Solution

Let's consider the first part of the problem: The angle between two straight lines is 45°, and the equation of one line is 4x - 6y = 1. We need to find the slope of the other line.

  1. Find the slope of the given line:

    • Rewrite the equation 4x - 6y = 1 in slope-intercept form (y = mx + c).
    • Subtract 4x from both sides: -6y = -4x + 1
    • Divide both sides by -6: y = (2/3)x - (1/6)
    • The slope of the given line (m1) is 2/3.

  2. Use the formula for the angle between two lines:

    • We are given that the angle θ is 45°, so tan 45° = 1.
    • Let m2 be the slope of the other line. We have:
      1 = |(2/3 - m2) / (1 + (2/3) * m2)|

  3. Solve for m2:

    • Remove the absolute value by considering both positive and negative cases:
      • Case 1: (2/3 - m2) / (1 + (2/3) * m2) = 1
      • Case 2: (2/3 - m2) / (1 + (2/3) * m2) = -1

  4. Solve Case 1:

    • Multiply both sides by (1 + (2/3) * m2): 2/3 - m2 = 1 + (2/3) * m2
    • Multiply all terms by 3 to eliminate fractions: 2 - 3m2 = 3 + 2m2
    • Combine like terms: -1 = 5m2
    • Solve for m2: m2 = -1/5

  5. Solve Case 2:

    • Multiply both sides by (1 + (2/3) * m2): 2/3 - m2 = -1 - (2/3) * m2
    • Multiply all terms by 3 to eliminate fractions: 2 - 3m2 = -3 - 2m2
    • Combine like terms: 5 = m2

  6. The slopes of the other line are -1/5 and 5:

    • There are two possible slopes for the other line, which makes sense because there are two lines that can form a 45° angle with the given line.

Key Takeaways

  • Understanding the relationship between the angle between two lines and their slopes is crucial for solving problems in coordinate geometry.
  • The formula tan θ = |(m1 - m2) / (1 + m1 * m2)| is a powerful tool for finding the slope of a line when the angle between the lines and the slope of one line are known.
  • Remember to consider both positive and negative cases when removing the absolute value in the formula.

Verifying Right Triangles Using Coordinate Geometry

The second part of this discussion focuses on how to verify whether a triangle formed by three given points is a right triangle. A right triangle is a triangle that has one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called legs. The Pythagorean theorem is a fundamental concept in geometry that relates the sides of a right triangle: a² + b² = c², where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse.

In coordinate geometry, we can use the distance formula and the concept of slopes to determine whether a triangle is a right triangle. The distance formula helps us calculate the lengths of the sides, and the slopes help us determine if two sides are perpendicular (forming a right angle).

Methods for Verifying Right Triangles

There are two primary methods to verify if a triangle is a right triangle given the coordinates of its vertices:

  1. Using the Pythagorean Theorem:

    • Calculate the lengths of all three sides of the triangle using the distance formula.
    • Check if the square of the longest side is equal to the sum of the squares of the other two sides. If it is, the triangle is a right triangle.

  2. Using Slopes:

    • Calculate the slopes of the lines formed by each pair of vertices.
    • If any two slopes are negative reciprocals of each other (i.e., their product is -1), then the lines are perpendicular, and the triangle is a right triangle.

Step-by-Step Solution

Let's apply these methods to the given problem: Show that the points (2, 1) and (-2, 3) are the vertices of a right triangle. Let the three points be A(2, 1), B(-2, 3), and we need a third point to form a triangle. Let's assume the third point is C(x, y) for now, and we will analyze the provided points first to exemplify the method and then discuss how to generally approach finding or verifying right triangles.

Assuming there was a typo and a third point was provided, such as C(-1, -1), let's show that the points A(2, 1), B(-2, 3), and C(-1, -1) are vertices of a right triangle.

Method 1: Using the Pythagorean Theorem

  1. Calculate the lengths of the sides:
    • The distance between two points (x1, y1) and (x2, y2) is given by the formula:

d = √((x2 - x1)² + (y2 - y1)²) ```

*   Length of AB:
    ```
    AB = √((-2 - 2)² + (3 - 1)²) = √((-4)² + (2)²) = √(16 + 4) = √20
    ```

*   Length of BC:
    ```
    BC = √((-1 - (-2))² + (-1 - 3)²) = √(1² + (-4)²) = √(1 + 16) = √17
    ```

*   Length of AC:
    ```
    AC = √((-1 - 2)² + (-1 - 1)²) = √((-3)² + (-2)²) = √(9 + 4) = √13
    ```
  1. Check the Pythagorean theorem:

    • Identify the longest side: AB = √20

    • Check if AB² = BC² + AC²

    • (√20)² = (√17)² + (√13)²

    • 20 = 17 + 13

    • 20 ≠ 30

    • It seems there may be an issue in the assumption of point C or the original points provided do not form a right triangle with C(-1, -1). This illustrates the process, but let's proceed with Method 2 to confirm and highlight the alternative approach.

Method 2: Using Slopes

  1. Calculate the slopes of the sides:
    • The slope of a line between two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1) ```

*   Slope of AB:
    ```
    mAB = (3 - 1) / (-2 - 2) = 2 / -4 = -1/2
    ```

*   Slope of BC:
    ```
    mBC = (-1 - 3) / (-1 - (-2)) = -4 / 1 = -4
    ```

*   Slope of AC:
    ```
    mAC = (-1 - 1) / (-1 - 2) = -2 / -3 = 2/3
    ```

  1. Check for perpendicular lines:

    • Check if the product of any two slopes is -1.
    • mAB * mBC = (-1/2) * (-4) = 2 ≠ -1
    • mAB * mAC = (-1/2) * (2/3) = -1/3 ≠ -1
    • mBC * mAC = (-4) * (2/3) = -8/3 ≠ -1

  2. Conclusion (Based on the assumed point C):

    • None of the products of the slopes are -1, so no two sides are perpendicular. Therefore, the points A(2, 1), B(-2, 3), and C(-1, -1) do not form a right triangle. This reiterates the previous conclusion from Method 1.

Addressing the Original Question Intention

Given the original question, the key intention is likely to demonstrate the process. The fact that a third point isn't provided initially suggests focusing on the method rather than expecting a definitive 'yes' for a right triangle. The important aspect is illustrating how slopes are used.

Had a third point been part of the original question, the methods above would definitively indicate if a right triangle is formed.

General Steps for Verifying Right Triangles Using Slopes

  1. Calculate the slopes of all three sides of the triangle.
  2. Check if any two slopes are negative reciprocals of each other. This means their product should be -1. If you find such a pair, the triangle is a right triangle.
  3. If no such pair is found, the triangle is not a right triangle.

Key Takeaways

  • The Pythagorean theorem and the concept of slopes are powerful tools for verifying right triangles in coordinate geometry.
  • The distance formula is used to calculate the lengths of the sides of the triangle.
  • If the square of the longest side equals the sum of the squares of the other two sides, the triangle is a right triangle.
  • If the product of the slopes of any two sides is -1, the triangle is a right triangle.
  • Understanding these methods is crucial for solving a variety of geometry problems.

Conclusion

In this comprehensive guide, we have explored two fundamental concepts in coordinate geometry: determining the slope of a line given the angle between it and another line, and verifying whether a triangle formed by three given points is a right triangle. We have discussed the underlying principles, provided step-by-step solutions, and illustrated these concepts with examples. By mastering these concepts, you will be well-equipped to tackle a wide range of problems in mathematics and related fields. Remember, practice is key to solidifying your understanding, so try applying these methods to various examples to enhance your skills.