Question

lesson 11 practice problems

1. which of these constructions would construct a line of reflection that takes the point a to point b?

a. construct the perpendicular bisector of segment ab.
b. construct a line through b perpendicular to segment ab.
c. construct the line passing through a and b.
d. construct a line parallel to line ab.

1. a point p stays in the same location when it is reflected over line ℓ.

what can you conclude about p?

1. lines ℓ and m are perpendicular with point of intersection p.

noah says that a 180 - degree rotation, with center p, has the same effect on points in the plane as reflecting over line m. do you agree with noah? explain your reasoning.

Explanation:

1.

Step1: Recall reflection property

The line of reflection that takes point $A$ to point $B$ is the perpendicular bisector of segment $AB$. This is because any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment, and reflection over this line will map one endpoint to the other.

Step1: Understand reflection concept

If a point $P$ stays in the same location when reflected over line $\ell$, then point $P$ lies on the line of reflection $\ell$. This is because the distance from $P$ to $\ell$ is $0$ in this case, and reflection preserves distances from the line of reflection.

Step1: Analyze rotation and reflection effects

Let's consider a point $Q$ in the plane. A $180 -$ degree rotation about point $P$ takes a point $Q$ to a point $Q'$ such that $P$ is the mid - point of segment $QQ'$. When we reflect a point $Q$ over line $m$ (where $m\perp\ell$ and the intersection is $P$), the result is different. For a non - special case, a $180 -$ degree rotation about $P$ and reflection over line $m$ do not have the same effect. Only for points on line $m$ will the two transformations have the same result, but for points not on line $m$, they are different. For example, take a point $Q$ not on line $m$ and $\ell$. After a $180 -$ degree rotation about $P$, the new point $Q'$ is on the opposite side of $P$ from $Q$. After reflection over line $m$, the new point is on the opposite side of line $m$ from $Q$ in a different way compared to the rotation result.

Answer:

A. Construct the perpendicular bisector of segment $AB$.

2.