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°.
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?
1 answer
1 answer