T I a transformation represented by the matrix (5x 2) under T a square whose area is 10cm³
(-3 x)
is mapped onto a square of 110cm².Find the possible values of x
To find the possible values of x, we need to determine the scale factor of the transformation.
The area of the square after the transformation is given as 110 cm².
The area of the original square is 10 cm³, but since it is a square, the area can also be expressed as s², where s is the length of a side of the square.
So, 10 cm² = s².
Taking the square root of both sides, we get:
√10 cm = s
Since the transformation multiplies the length of the side by a scale factor of 5, the length of the side after the transformation is 5√10 cm.
Setting this length equal to the length of the side of the transformed square (x):
5√10 cm = x
Therefore, the possible values of x are 5√10 cm.
To find the possible values of x, we need to determine the scale factor of the transformation T.
The area of a square is calculated by squaring the length of one side. Let's assume the length of the side of the square before the transformation is "a" cm. Therefore, the area of the original square is a² cm².
After the transformation T, the area of the new square is given as 110 cm². Let's label the length of the side of the new square as "b" cm. Therefore, the area of the new square is b² cm².
According to the given information, T is represented by the matrix (5x, 2) under T, where x is the length of the side of the original square.
Now, let's use the formula for finding the scale factor (k) of a transformation using the matrix representation:
k = sqrt(determinant of the matrix representing the transformation)
The matrix representing the transformation T is:
[5x]
[2]
The determinant of this matrix is calculated as follows:
5x * 2 - (-3) * 2
= 10x + 6
= 10(x + 0.6)
Therefore, the scale factor (k) of the transformation T is:
k = sqrt(10(x + 0.6))
Since the original square has an area of 10 cm³, we can set up the following equation for the scale factor:
a² = 10
a = sqrt(10)
Now, we can substitute a into the equation for k:
k = sqrt(10(sqrt(10) + 0.6))
By equating the scale factors of the two squares, we can solve for x:
k = sqrt(10(sqrt(10) + 0.6))
sqrt(10) = sqrt(10(sqrt(10) + 0.6))
10 = 10(sqrt(10) + 0.6)
Now, we can solve for sqrt(10) + 0.6:
sqrt(10) + 0.6 = 1
Finally, we can solve for x using the equation for the scale factor:
k = sqrt(10(x + 0.6))
1 = sqrt(10(x + 0.6))
1 = sqrt(10x + 6)
1^2 = 10x + 6
1 = 10x + 6
10x = -5
x = -0.5
Therefore, the possible value of x is -0.5.