Question

If AB=18 inches then find the distance of A′B′ after a reflection over the x-axis and a rotation of 226° clockwise.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the distance of A'B' after reflection over the x-axis and a rotation of 226° clockwise, we first need to reflect point A and B over the x-axis to get A' and B'.

After reflection over the x-axis, the coordinates of A' will be the same as A but with the y-coordinate negated, and the coordinates of B' will be the same as B but with the y-coordinate negated.

Let's assume the coordinates of A are (x, y) and the coordinates of B are (x', y'). Therefore, the coordinates of A' will be (x, -y) and the coordinates of B' will be (x', -y').

Now, we need to rotate A' and B' 226° clockwise. To rotate a point (x, y) about the origin in the clockwise direction, we can use the following formulas:
x' = x*cos(theta) + y*sin(theta)
y' = -x*sin(theta) + y*cos(theta)

where theta is the angle of rotation in radians.

In this case, theta = 226° = 226 * π / 180 ≈ 3.945 radians.

For point A':
x' = x*cos(3.945) + (-y)*sin(3.945)
y' = -x*sin(3.945) + (-y)*cos(3.945)

For point B':
x'' = x'*cos(3.945) + (-y')*sin(3.945)
y'' = -x'*sin(3.945) + (-y')*cos(3.945)

Now, we can substitute the values of x, y, x', and y' and calculate the coordinates of A'' and B''. Then, we can find the distance between them using the distance formula:
distance = sqrt((x'' - x')² + (y'' - y')²)

Note: The calculations involved may be complex and time-consuming, but knowing the formulas and following the steps will give you the desired result.

Answer 2

what's the inches of A'B'

Answer 3

[AI-generated answer — Bot GPT 3.5]

Given that AB has a length of 18 inches, we need to find the coordinates of A and B before we can reflect them over the x-axis and rotate them 226° clockwise.

Let's assume point A is at (0, 0) for simplicity, and hence B is at (18, 0) because AB has a length of 18 inches.

After reflecting A and B over the x-axis, we get A' at (0, 0) and B' at (18, 0) since reflecting over the x-axis doesn't change the x-coordinate.

Now, to rotate A' and B' by 226° clockwise, we will use the rotation formulas mentioned earlier:

For point A' (0, 0):
x' = 0*cos(3.945) + 0*sin(3.945) = 0
y' = -0*sin(3.945) + 0*cos(3.945) = 0

For point B' (18, 0):
x'' = 18*cos(3.945) + 0*sin(3.945) ≈ -16.67
y'' = -18*sin(3.945) + 0*cos(3.945) ≈ -7.09

Therefore, the coordinates of A'B' after reflection over the x-axis and rotation by 226° clockwise are (0, 0) and (-16.67, -7.09).

To find the distance between A'B', we use the distance formula:
distance = sqrt((-16.67 - 0)² + (-7.09 - 0)²)
distance ≈ sqrt(278.9 + 50.33)
distance ≈ sqrt(329.23)
distance ≈ 18.14 inches

Thus, the distance between A'B' after reflection over the x-axis and a rotation of 226° clockwise is approximately 18.14 inches.

Answer 4

What is point A (9, 7) after a translation of 1 units left and 1 units down followed by a 180° clockwise rotation around the origin.

Answer 5

[AI-generated answer — Bot GPT 3.5]

Let's break down the given transformations step by step:

1. Translation of 1 unit left and 1 unit down:
After the translation of 1 unit left (subtracting 1 from the x-coordinate) and 1 unit down (subtracting 1 from the y-coordinate), the new coordinates of point A(9, 7) will be:
A'(x', y') = (9-1, 7-1) = (8, 6)

2. 180° clockwise rotation around the origin:
To perform a 180° clockwise rotation about the origin, we use the following rotation formulas:
x' = x*cos(180°) + y*sin(180°)
y' = -x*sin(180°) + y*cos(180°)

Converting degrees to radians: 180° = π radians

For point A' (8, 6):
x'' = 8*cos(π) + 6*sin(π) = 8*(-1) + 6*0 = -8
y'' = -8*sin(π) + 6*cos(π) = -8*0 + 6*(-1) = -6

Therefore, after the translation of 1 unit left and 1 unit down followed by a 180° clockwise rotation, point A(9, 7) ends up at:
A''(x'', y'') = (-8, -6).

So, point A(9, 7) after a translation of 1 unit left and 1 unit down followed by a 180° clockwise rotation around the origin is A''(-8, -6).

Answer 6

The diameter of a circle was 6 cm and the diameter of the final circle was 48 cm. What is the scale factor?

Answer 7

[AI-generated answer — Bot GPT 3.5]

The scale factor is a ratio between the size of an object in one image or diagram to the size of that same object in another image or diagram. In this case, we are comparing the diameters of the initial circle and the final circle.

The scale factor can be determined by dividing the diameter of the final circle by the diameter of the initial circle. Let's calculate the scale factor:

Initial circle diameter = 6 cm
Final circle diameter = 48 cm

Scale Factor = Final Circle Diameter / Initial Circle Diameter
Scale Factor = 48 cm / 6 cm
Scale Factor = 8

Therefore, the scale factor between the initial circle and the final circle is 8. This means that the final circle is 8 times larger than the initial circle.

Answer 8

The hypotenuse of a triangle was 2 cm and the hypotenuse of the final triangle was 1.3 cm. What is the scale factor?

Answer 9

[AI-generated answer — Bot GPT 3.5]

In a right triangle, the hypotenuse is the side opposite the right angle and is the longest side of the triangle. The scale factor is the ratio of the lengths of corresponding sides or measurements of two similar figures.

Given:
Initial triangle hypotenuse: 2 cm
Final triangle hypotenuse: 1.3 cm

To find the scale factor, we compare the lengths of the hypotenuses of the two triangles:

Scale Factor = Final Triangle Hypotenuse / Initial Triangle Hypotenuse
Scale Factor = 1.3 cm / 2 cm
Scale Factor = 0.65

Therefore, the scale factor between the initial triangle and the final triangle is 0.65. This means that the final triangle is 0.65 times (or 65% of) the size of the initial triangle.