Determinants - Examples on Area of triangles

  • In this lesson, we will learn how to calculate the area of triangles using determinants.
  • Determinant is a mathematical concept used to solve a system of linear equations.
  • Determinant of a matrix is denoted by |A| or det(A), where A is the given matrix.
  • The formula to calculate the area of a triangle using determinants is:
    • Area = 1/2 * determinant of matrix Example 1: Find the area of a triangle with vertices at (1, 2), (3, 4), and (5, 6). Solution: Let’s first create a matrix using the given coordinates: A = | 1 2 | | 3 4 | | 5 6 | Now, calculate the determinant of A: det(A) = (1 * 4) - (3 * 2) = 4 - 6 = -2 Finally, plug the determinant value into the area formula: Area = 1/2 * (-2) = -1 Therefore, the area of the triangle is -1 square units. Example 2: Find the area of a triangle with vertices at (-2, 1), (4, -3), and (5, 2). Solution: Let’s create the matrix A using the given coordinates: A = | -2 1 | | 4 -3 | | 5 2 | Calculate the determinant of matrix A: det(A) = (-2 * -3) - (1 * 4) = 6 - 4 = 2 Using the area formula, we have: Area = 1/2 * 2 = 1 Hence, the area of the triangle is 1 square unit. Remember: The area of a triangle cannot be negative. Example 3: Let’s consider a triangle with vertices at (-1, 0), (2, 5), and (4, 1). Solution: Construct the matrix A using the given coordinates: A = | -1 0 | | 2 5 | | 4 1 | Find the determinant of matrix A: det(A) = (-1 * 5) - (0 * 2) = -5 Using the area formula, we get: Area = 1/2 * (-5) = -2.5 Thus, the area of the triangle is -2.5 square units.
  1. Determinants - Examples on Area of triangles
  • In this lesson, we will continue exploring examples on calculating the area of triangles using determinants.
  • Determinants are a powerful tool for solving systems of linear equations and finding the area of geometric shapes.
  • The formula to calculate the area of a triangle using determinants is: Area = 1/2 * determinant of matrix
  1. Example 4:
  • Find the area of a triangle with vertices at (0, 0), (3, 4), and (6, 0).
  • Matrix A: | 0 0 | | 3 4 | | 6 0 |
  • Calculate the determinant of A: det(A) = (0 * 4) - (0 * 3) = 0
  • Using the area formula, we have: Area = 1/2 * 0 = 0
  • Therefore, the area of the triangle is 0 square units, which indicates that the points are collinear.
  1. Example 5:
  • Consider a triangle with vertices at (2, 1), (-3, 5), and (1, 6).
  • Matrix A: | 2 1 | | -3 5 | | 1 6 |
  • Calculate the determinant of A: det(A) = (2 * 5) - (1 * -3) = 7
  • Using the area formula, we get: Area = 1/2 * 7 = 3.5
  • Thus, the area of the triangle is 3.5 square units.
  1. Example 6:
  • Find the area of a triangle with vertices at (-4, -5), (-2, 3), and (0, -1).
  • Matrix A: | -4 -5 | | -2 3 | | 0 -1 |
  • Calculate the determinant of A: det(A) = (-4 * 3) - (-5 * -2) = -2
  • Using the area formula, we have: Area = 1/2 * (-2) = -1
  • The negative area indicates that the triangle is oriented clockwise.
  • Hence, the area of the triangle is -1 square unit.
  1. Alternate Formula for Area:
  • While we have been using the determinant formula, there is an alternate formula for finding the area of a triangle using coordinates.
  • Given points (x₁, y₁), (x₂, y₂), and (x₃, y₃), the area can be calculated as follows:
    • Area = 1/2 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
  1. Example 7:
  • Find the area of a triangle with vertices at (1, 2), (-3, 5), and (5, 6) using the alternate formula.
  • Area = 1/2 * |1(5-6) - (-3)(6-2) + 5(2-5)| = 1/2 * |-1 - (-12) + 15| = 1/2 * |-1 + 12 + 15| = 1/2 * 26 = 13
  • Therefore, the area of the triangle is 13 square units.
  1. Example 8:
  • Calculate the area of a triangle with vertices at (-1, 4), (3, -2), and (7, 1) using the alternate formula.
  • Area = 1/2 * |-1(-2-1) + 3(1-4) + 7(4-(-2))| = 1/2 * |-1 - 9 + 42| = 1/2 * 32 = 16
  • Hence, the area of the triangle is 16 square units.
  1. Properties of Triangle Area:
  • The area of a triangle is always positive, irrespective of the orientation of the points.
  • The area is zero if the points are collinear.
  • The area remains the same when the triangle is translated, rotated, or scaled.
  1. Summary:
  • Determinants help calculate the area of triangles.
  • The formula for calculating the area using determinants is: Area = 1/2 * determinant of matrix.
  • The alternate formula uses coordinates and is: Area = 1/2 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
  • The area of a triangle is always positive, except when it’s zero for collinear points.
  • The area remains unchanged under transformations.
  1. Practice Questions
  • Find the area of a triangle with vertices at (6, 3), (2, 7), and (-1, -4).
  • Calculate the area of a triangle with vertices at (-2, 3), (3, -5), and (4, 1).
  • Using determinants, find the area of a triangle with vertices at (0, 0), (5, 0), and (3, 2).
  • Find the area of a triangle with vertices at (4, 2), (1, 3), and (-2, 0) using the alternate formula.
  • Calculate the area of a triangle with vertices at (-4, -1), (3, 2), and (0, -3) using determinants.
  1. Example 9:
  • Find the area of a triangle with vertices at (2, -3), (-5, 4), and (3, 6) using the alternate formula.
  • Area = 1/2 * |2(4-6) + (-5)(6-(-3)) + 3((-3)-4)| = 1/2 * |-4 - (-45) + (-21)| = 1/2 * 20 = 10
  • Hence, the area of the triangle is 10 square units.
  1. Example 10:
  • Calculate the area of a triangle with vertices at (-3, 2), (1, -1), and (-4, 5) using the alternate formula.
  • Area = 1/2 * |-3(-1-5) + 1(5-2) + (-4)(2-(-1))| = 1/2 * |18 + 3 + (-6)| = 1/2 * 15 = 7.5
  • Therefore, the area of the triangle is 7.5 square units.
  1. Example 11:
  • Find the area of a triangle with vertices at (0, 0), (6, 0), and (4, 3) using determinants.
  • Matrix A: | 0 0 | | 6 0 | | 4 3 |
  • Calculate the determinant of A: det(A) = (0 * 0) - (0 * 6) = 0
  • Using the area formula, we have: Area = 1/2 * 0 = 0
  • Thus, the area of the triangle is 0 square units, indicating that the points are collinear.
  1. Example 12:
  • Calculate the area of a triangle with vertices at (-4, -1), (3, 2), and (0, -3) using determinants.
  • Matrix A: | -4 -1 | | 3 2 | | 0 -3 |
  • Calculate the determinant of A: det(A) = (-4 * 2) - (-1 * 3) = -5
  • Using the area formula, we have: Area = 1/2 * (-5) = -2.5
  • Hence, the area of the triangle is -2.5 square units.
  1. Summary:
  • Determinants can be used to find the area of triangles.
  • The area can be calculated using the determinant formula: Area = 1/2 * determinant of matrix.
  • Alternatively, the area can be found using the coordinates: Area = 1/2 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|.
  • The area is always positive, except when it’s zero for collinear points.
  • Transformations such as translation, rotation, or scaling do not change the area.
  1. Practice Questions
  • Find the area of a triangle with vertices at (3, 4), (-2, -5), and (7, 0) using the alternate formula.
  • Calculate the area of a triangle with vertices at (0, -2), (4, 1), and (-1, 5) using the alternate formula.
  • Using determinants, find the area of a triangle with vertices at (2, 0), (-1, 3), and (0, -1).
  • Find the area of a triangle with vertices at (-5, 2), (1, -3), and (3, 4) using the alternate formula.
  • Calculate the area of a triangle with vertices at (-2, 0), (4, 3), and (6, 2) using determinants.
  1. Formulas for Area:
  • Determinants: Area = 1/2 * determinant of matrix
  • Alternate formula: Area = 1/2 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
  • Triangle area cannot be negative, except when the points are collinear.
  • Area remains unchanged under transformations.
  1. Example 13:
  • Find the area of a triangle with vertices at (1, 1), (4, -2), and (7, 3) using the alternate formula.
  • Area = 1/2 * |1(-2-3) + 4(3-1) + 7(1-(-2))| = 1/2 * |-2 + 8 + 21| = 1/2 * 27 = 13.5
  • Hence, the area of the triangle is 13.5 square units.
  1. Example 14:
  • Calculate the area of a triangle with vertices at (0, -4), (3, 0), and (-1, 2) using the alternate formula.
  • Area = 1/2 * |0(0-2) + 3(2+4) + (-1)(-4-0)| = 1/2 * |0 + 18 + 4| = 1/2 * 22 = 11
  • Therefore, the area of the triangle is 11 square units.
  1. Recap:
  • Determinants can be used to calculate the area of triangles.
  • The formula using determinants is: Area = 1/2 * determinant of matrix.
  • Alternatively, the area can be found using the coordinates with the alternate formula.
  • Area is always positive, except when it’s 0 for collinear points.
  • Transformations do not change the area of a triangle.

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