The figure shows a 10 x 10 square centered at the origin. The darker shaded square is the image of this square, dilated by a factor of 0.8 and rotated about the origin by exactly the number of degrees, theta, so that its corners touch the sides of the parent square. The corners of the rotated square divide the sides of the 10 x 10 squade into segments of length a and b. Use the fact that a and b add up to 10 to find the value of theta.
To find the value of theta, we can use the information given about the segments a and b. Let's start by understanding the problem.
From the figure, it is known that the corners of the rotated square touch the sides of the 10 x 10 square. Let's assume that the initial square has its vertices at points A(-5, 5), B(5, 5), C(5, -5), and D(-5, -5). The side length of this initial square is 10.
Now, let's find the coordinates of the corners of the rotated square. Since the rotated square is dilated by a factor of 0.8, its side length becomes (0.8) * 10 = 8.
To rotate the square such that its corners touch the sides of the parent square, we need to find the coordinates of the corners of the rotated square.
Let's start with the top-right corner of the rotated square (let's call it point E). Since the rotated square is centered at the origin, the coordinates of E will be (4, 4).
Similarly, we can find the coordinates of the other corners:
- Top-left corner (let's call it F): (-4, 4)
- Bottom-left corner (let's call it G): (-4, -4)
- Bottom-right corner (let's call it H): (4, -4)
Now, we can find the lengths of the segments a and b using the coordinates of the corners.
Segment a is the length of the horizontal distance between points B and C. As the x-coordinates of both points are 5, this length is given by:
a = 5 - (-5) = 10
Segment b is the length of the vertical distance between points A and B. As the y-coordinates of both points are 5, this length is given by:
b = 5 - (-5) = 10
According to the information given, a and b should add up to 10. So, we have:
a + b = 10
Substituting the values of a and b, we get:
10 + 10 = 10
This equation is not true, suggesting that our initial assumption for the rotation angle theta is incorrect.
To find the value of theta, we need to adjust our initial assumption. The sides of the 10 x 10 square should be divided into segments of length a and b using the rotated square's corners, not the initial square's corners.
Let's analyze the sides of the rotated square to find the correct a and b values.
From the figure, it is evident that:
- The horizontal distance between E and H is equal to a.
- The vertical distance between E and F is equal to b.
Looking at the coordinates of E and H, we can calculate the horizontal distance:
a = 4 - (-4) = 8
Similarly, we can calculate the vertical distance using the coordinates of E and F:
b = 4 - 4 = 0
Now, let's check if a + b = 10:
8 + 0 = 8
Since 8 + 0 is not equal to 10, our assumption for theta is still incorrect.
To find the correct value of theta, we need to consider the fact that the rotated square is dilated by a factor of 0.8. This means that the side length of the rotated square is 0.8 times smaller than the side length of the initial square.
Let's adjust our calculation for the lengths of a and b using this information.
Since the initial square has a side length of 10, the side length of the rotated square is (0.8) * 10 = 8.
Now, let's recalculate the lengths of a and b using the scaled coordinates of the corners:
For horizontal lengths (a), we need to consider the x-coordinates of E and H:
a = 4 * 0.8 - (-4 * 0.8) = 6.4 + 3.2 = 9.6
For vertical lengths (b), we need to consider the y-coordinates of E and F:
b = 4 - (-4) = 8
Now, let's check if a + b = 10:
9.6 + 8 = 17.6
Since 9.6 + 8 does not equal 10, our current assumption for theta is also incorrect.
To find the correct value of theta, we need to adjust our approach. Instead of directly calculating the lengths of a and b using the coordinates of the corners, we should use the trigonometric properties of the rotated square.
By observing the figure, we can see that theta represents the angle of rotation. The tangent of this angle can be determined by analyzing the right triangle formed between points E, F, and the origin.
The side opposite to theta is the vertical length b, and the side adjacent to theta is half of the horizontal length a since the rotated square is centered at the origin.
Let's express the tangent of theta using the values of b and half of a (a/2):
tan(theta) = b / (a/2)
Substituting the known values of b = 8 and a/2 = 9.6/2 = 4.8, the equation becomes:
tan(theta) = 8 / 4.8
To find the value of theta, we can take the inverse tangent (arctan) of both sides:
theta = arctan(8 / 4.8)
Using a calculator, we find:
theta ≈ 60.26 degrees (rounded to two decimal places)
Therefore, the approximate value of theta is 60.26 degrees.