Question

.Find the measure of the acute or right angle formed by intersecting lines so that C can be mapped to C′ using two reflections. A rotation (x,y) → (-x,-y) maps C to C′. The measure of the angle is ____

Answer 1

The angle between intersecting lines that would allow two reflections to map C to C' is 90 degrees, assuming a rotation of 180 degrees around the origin.

Step-by-step explanation:

The student is concerned with finding the measure of an acute or right angle formed by two intersecting lines, where a rotation maps point C to C'. Given that the rotation is (x, y) → (-x, -y), this indicates a 180-degree rotation around the origin. For two reflections to result in a 180-degree rotation, each reflection must be over a line that is at a 45-degree angle to the original orientation. Therefore, the angle between the intersecting lines that would allow two reflections to map C to C' is 90 degrees.

Answer 2

The measure of the angle formed by the two intersecting lines for the two reflections is 90 degrees, as this would create a 180-degree rotation to map point C to C'.

Step-by-step explanation:

The measure of the angle formed by the two intersecting lines that would allow point C to be mapped to C' through two reflections can be found by assessing the rotation given. The rotation (x,y) → (-x,-y) represents a 180-degree rotation. This is because if we take a point (x,y) and rotate it 180 degrees around the origin, we end up at (-x,-y). Therefore, to map C to C' by using two reflections, we need to find two lines that intersect at an angle that would lead to a 180-degree rotation when reflected across both.

Using the law of reflection, which states the angle of reflection equals the angle of incidence, we can deduce that for two reflections to result in a 180-degree rotation, each reflection must involve a 90-degree angle with the respective line of reflection. Thus, the first reflection would map C to a temporary point, say C'', and the second reflection would map C'' to C'. Both reflections are at 90 degrees to their respective lines. Therefore, the lines must be perpendicular to each other, and the angle between them would be 90 degrees.