Chapter 11, Problem 1E

In $\underset{\_}{\text{Fig}\text{.}11-35}$, indicate which point is the image of $P$ under

a. The reflection with axis $l_1$.

b. The reflection with axis $l_2$.

c. The reflection with axis $l_3$.

d. The reflection with axis $l_4$.

$Figure11-35$

To determine

1.

To find:

The image of $P$ under the reflection of axis $l_1$.

Answer to Problem 1E

Solution:

The image of point P under the reflection with axis $l_1$ is point C.

Explanation of Solution

Given:

The given figure is,

A rigid motion that moves the object from its starting position to a new position, namely the mirror image of the starting position is called a reflection in the plane.

If a point $Q$ has an image $Q^{\prime }$ under reflection, then the axis of reflection is the perpendicular bisector of the line segment $Q{Q}^{\prime }$.

To find a image of point $P$. Draw a perpendicular line that pass through $P$ namely $l$. The image of $P$ is the point $P^{\prime }$ that lies on $l$ and has the same distance from $l$ as that of point $P$.

Consider the given figure,

Figure (1)

In the given figure, the points E, C, and D lie on the opposite side of $l_1$.

Points E, D are opposite to point P but do not have same distance from $l_1$ as that of P.

The point C is at same distance from $l_1$ as that of point P.

The image of P is point C.

Conclusion:

Thus, the image of point P under the reflection with axis $l_1$ is point C.

To determine

(b)

To find:

The image of $P$ under the reflection of axis $l_2$.

Answer to Problem 1E

Solution:

The image of point P under the reflection with axis $l_2$ is point F.

Explanation of Solution

Given:

The given figure is,

A rigid motion that moves the object from its starting position to a new position, namely the mirror image of the starting position is called a reflection in the plane.

If a point $Q$ has an image $Q^{\prime }$ under reflection, then the axis of reflection is the perpendicular bisector of the line segment $Q{Q}^{\prime }$.

To find a image of point $P$. Draw a perpendicular line that pass through $P$ namely $l$. The image of $P$ is the point $P^{\prime }$ that lies on $l$ and has the same distance from $l$ as that of point $P$.

Consider the given figure,

Figure (2)

In Figure (1), the points E, F, and D lie on the opposite side of $l_2$.

Points E, D are opposite to point P but do not have same distance from $l_2$ as that of P.

The point F is at same distance from $l_2$ as that of Point P.

The image of P is point F.

Conclusion:

Thus, the image of point P under the reflection with axis $l_2$ is point F.

To determine

(c)

To find:

The image of $P$ under the reflection of axis $l_3$.

Answer to Problem 1E

Solution:

The image of $P$ under the reflection of axis $l_3$ is E.

Explanation of Solution

Given:

The given figure is,

A rigid motion that moves the object from its starting position to a new position, namely the mirror image of the starting position is called a reflection in the plane.

If a point $Q$ has an image $Q^{\prime }$ under reflection, then the axis of reflection is the perpendicular bisector of the line segment $Q{Q}^{\prime }$.

To find a image of point $P$. Draw a perpendicular line that pass through $P$ namely $l$. The image of $P$ is the point $P^{\prime }$ that lies on $l$ and has the same distance from $l$ as that of point $P$.

Consider the given figure,

Figure (3)

In the given figure, the points C, D, and E lie on the opposite side of $l_3$.

Points C and D are opposite to point P but do not have same distance from $l_3$ as that of P.

Point E has same distance from $l_3$ as that of point P.

The image of P is point E.

Conclusion:

Thus, the image of point P under the reflection with axis $l_3$ is point E.

To determine

(d)

To find:

The image of $P$ under the reflection of axis $l_4$.

Answer to Problem 1E

Solution:

The image of $P$ under the reflection of axis $l_4$ is B.

Explanation of Solution

Given:

The given figure is,

A rigid motion that moves the object from its starting position to a new position, namely the mirror image of the starting position is called a reflection in the plane.

If a point $Q$ has an image $Q^{\prime }$ under reflection, then the axis of reflection is the perpendicular bisector of the line segment $Q{Q}^{\prime }$.

To find a image of point $P$. Draw a perpendicular line that pass through $P$ namely $l$. The image of $P$ is the point $P^{\prime }$ that lies on $l$ and has the same distance from $l$ as that of point $P$.

Consider the given figure,

Figure (4)

In the given figure, the points A, B, C, D, and G lie on the opposite side of $l_4$.

Points A, D, C and G are opposite to point P but do not have same distance from $l_4$ as that of P.

Distance of point B from $l_4$ is same as that of point P.

The image of P is point B.

Conclusion:

Thus, the image of point P under the reflection with axis $l_4$ is point B.