Rectangle LMNO is shown below.
If rectangle LMNO were rotated 180° clockwise about the origin to create rectangle L'M'N'O', what would the measure of angle O' be?
When a point is rotated 180° about the origin, its coordinates change to their opposites.
The coordinates of point O are (2,5). When rotated 180° about the origin, the x-coordinate becomes -2 and the y-coordinate becomes -5. Therefore, point O' is located at (-2,-5).
To find the measure of angle O', we need to use the slope formula. The slope of the diagonal LN is given by:
slope of LN = change in y / change in x
slope of LN = (5 - 0) / (2 - 0)
slope of LN = 5/2
Since L'N' is perpendicular to LN, its slope is the negative reciprocal of the slope of LN. Therefore, the slope of L'N' is -2/5.
Since L'N' passes through point O', we can use the slope-intercept form of a line (y = mx + b) to find the equation of L'N'. Substituting the coordinates of point O' into the equation, we get:
-5 = (-2/5)(-2) + b
-5 = 4/5 + b
-5 - 4/5 = b
-25/5 - 4/5 = b
-29/5 = b
Therefore, the equation of L'N' is y = -2/5x - 29/5.
To find the measure of angle O', we need to find the slope of the diagonal O'N'. The slope of a line is given by the coefficient of x when the equation is in the form y = mx + b. The equation of L'N' is already in this form, so the slope of O'N' is -2/5.
The measure of an angle is equal to the arctangent of the slope. Therefore, the measure of angle O' can be found using:
angle O' = arctan(-2/5)
Using a calculator, we find that angle O' is approximately -21.8° (rounded to the nearest tenth).
Therefore, the measure of angle O' is approximately -21.8°.
To find the measure of angle O', we need to understand how the rotation of 180° clockwise affects the angles of the rectangle.
1. In the original rectangle LMNO, angle O is a right angle (90°).
2. When a point is rotated 180° clockwise about the origin, its coordinates change to (-x, -y).
3. Since angle O is a right angle (90°), rotating it 180° clockwise will mean the new angle O' will also be a right angle.
Therefore, the measure of angle O' in rectangle L'M'N'O' will also be 90°.
To find the measure of angle O' after rotating rectangle LMNO 180° clockwise about the origin, we can use the concept of rotational symmetry.
Here's how you can determine the measure of angle O':
1. Visualize rectangle LMNO in the Cartesian coordinate plane with the origin O at (0,0) and the four vertices L, M, N, and O labeled accordingly.
2. To rotate a point (x, y) 180° clockwise about the origin, we need to apply the following transformation:
- The x-coordinate becomes -x
- The y-coordinate becomes -y
3. Apply this transformation to each of the rectangle's vertices (L, M, N, and O) to obtain the new coordinates of L', M', N', and O'. The new rectangle L'M'N'O' will be an image of the original rectangle LMNO after the rotation.
4. Once you have the coordinates of O', you can determine the measure of angle O' using trigonometry or the properties of the coordinate plane.
Keep in mind that the exact measure of angle O' will depend on the dimensions and orientation of rectangle LMNO. By following the steps outlined above and considering the specific coordinates of the vertices, you can calculate the measure of angle O' accurately.