Triangle ABC with an area of 15cm3 is mapped onto triangle A'B'C' using matrix M=(2 -3)
(1 1).Find the area of the triangle A'B'C'
To find the area of triangle A'B'C', we can use the formula for the area of a triangle given its vertices. However, since the matrix M represents a linear transformation, the area of the triangle A'B'C' will be scaled by the determinant of the matrix.
The given matrix is M = (2 -3)
(1 1)
The determinant of M is det(M) = (2*1) - (-3*1) = 2 + 3 = 5.
The area of triangle A'B'C' can be calculated as follows:
Area(A'B'C') = det(M) * Area(ABC)
= 5 * 15 cm^2
= 75 cm^2
Therefore, the area of triangle A'B'C' is 75 cm^2.
To find the area of triangle A'B'C', we can use the matrix transformation given by matrix M.
The matrix M represents a linear transformation that maps the vertices of triangle ABC to the vertices of triangle A'B'C'.
The formula to find the area of a triangle is given by the formula: Area = (1/2) * base * height.
In our case, the base and height of triangle ABC will be transformed using the matrix M to find the corresponding base and height for triangle A'B'C'.
To find the new base and height, we can multiply the matrix M by the corresponding base and height vectors of triangle ABC.
Let's call the base and height vectors of triangle ABC as v_base and v_height respectively.
v_base = vector [base_of_ABC, 0]
v_height = vector [0, height_of_ABC]
Multiplying matrix M by the base and height vectors:
M * v_base = [2 -3] * vector [base_of_ABC, 0] = [2*base_of_ABC - 3*0] = [2*base_of_ABC]
M * v_height = [2 -3] * vector [0, height_of_ABC] = [2*0 - 3*height_of_ABC] = [-3*height_of_ABC]
So, the new base and height vectors for triangle A'B'C' are [2*base_of_ABC, -3*height_of_ABC].
Now, the area of triangle A'B'C' can be calculated using the transformed base and height vectors:
Area of A'B'C' = (1/2) * new_base * new_height
= (1/2) * (2*base_of_ABC) * (-3*height_of_ABC)
= -3 * base_of_ABC * height_of_ABC
Given that the area of triangle ABC is 15 cm^2, we can use the formula to find the area of triangle A'B'C':
15 = -3 * base_of_ABC * height_of_ABC
To solve for base_of_ABC and height_of_ABC, we can rearrange the equation:
base_of_ABC * height_of_ABC = -15/3
base_of_ABC * height_of_ABC = -5
Therefore, the area of triangle A'B'C' is -5 cm^2.