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# Reflection, Math Assignment Help

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Geometry Mathematics - Reflection, Math

** Reflection**

**Reflection:** The process of obtaining its image in the plane mirror is called reflection.

If m is the plane mirror, then

(i) P' is the image of P under reflection in M and NP = NP'.

(ii) NP' is the image of NP under reflection.

(iii) PP' is perpendicular to M and M is perpendicular bisector or mediator of PP'.

(Mediator is another name for perpendicular bisector).

**(A) Point reflection:** The reflection of A through the point 0 is the point A' such that O is the mid-point of AA'. The point O is called centre of reflection.

** **

**INVARIANT POINTS, LINES AND LINE SEGMENTS **

(B) **Reflection in x-axis:** To get the image of a point after its reflection in the x-axis, we change the sign of the x coordinate but keep the y co-ordinate the same

i.e., M_{y} (x, y) = (- x, y)

(i) Under reflection in the x-axis, all points on the x-axis are invariant i.e., they are their own image.

Under reflection in the x-axis, lines invariant are:

(a) the x-axis,

(b) the line perpendicular to the x-axis: Line segments right bisected by the x-axis are invariant under reflection in' the x-axis,

(C) **Reflection in y-axis:** To find the image of any point after reflection in the y-axis, we should retain the x co-ordinate and change the sign of the y co-ordinate.

i.e., Mx (x, y) = (x, - y)

(ii) Under reflection in the y-axis, all points on the y-axis are invariant.

Under reflection in the y-axis, lines invariant are:

(a) the y-axis.

(b) All lines perpendicular to the y-axis (parallel to x-axis): Line segments right bisected by the y-axis are invariant under reflection in the origin.

(D) **Reflection in the origin:** To get the image of a point after its reflection in the origin, we change the signs of both x co-ordinate and y co-ordinate.

(iii) The only point, invariant under reflection in the origin is the origin itself. r_{m} denote the operation 'reflection in m' and is written 'as r_{m}: P → P' or r_{m} (P) = P'.

The point P is called the pre-image of the point P'.** **

**Note:**

(1) x-co-ordinate or abscissa of all the points on the y-axis is zero. Thus, any point on the y-axis is of the form (0. y).

(2) y-co-ordiante or ordinate of all the points on the x-axis, is zero. Thus, any points on the x-axis is of the form (x, 0).

(3) Co-ordinates of origin are (0, 0) as t ms on both the axes.

**RFLECTION CONSTRUCTION: **

**Reflection of a point in a line: **

**Steps of Construction: **

1. Draw PQ perpendicular to AB, meeting AB at O.

2. From OQ cut of OP' = OP.

Then P' is the reflection of P in AB.

**REFLECTION OF A LINE SEGMENT IN A LINE**

**Steps of Construction: **

1. Draw PO perpendicular to AB.

Produce PO and cut of OP' = PO.

\ P' is the reflection of P in AB.

2. Similarly, get a', the image of a in AB.

3. Join P' and Q'.

Then P'O' is the reflection of line segment PQ in AB.

** **

**Reflection of a point in a point:** Let o be the point in which we require to find the reflection of point P.

** **

**Steps of Construction: **

Join PO and produce it to P' such that PO = OP'

Then P' is the reflection of P in the point O.

**Reflection of a line segment in a point:** Let O be the point in which we require to find the reflection of line segment PQ.

** **

**Steps of Construction: **

1. Join PO and produce to P' such that PO = OP'.

2. Join QO and produce to Q, such that OO = OQ.

3. Join P' and Q'.

Then P'Q' is the reflection of line segment PQ in the point O.

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**Reflection**

**Reflection:**The process of obtaining its image in the plane mirror is called reflection.

**(A) Point reflection:**The reflection of A through the point 0 is the point A' such that O is the mid-point of AA'. The point O is called centre of reflection.

**INVARIANT POINTS, LINES AND LINE SEGMENTS**

**Reflection in x-axis:**To get the image of a point after its reflection in the x-axis, we change the sign of the x coordinate but keep the y co-ordinate the same

_{y}(x, y) = (- x, y)

**Reflection in y-axis:**To find the image of any point after reflection in the y-axis, we should retain the x co-ordinate and change the sign of the y co-ordinate.

**Reflection in the origin:**To get the image of a point after its reflection in the origin, we change the signs of both x co-ordinate and y co-ordinate.

_{m}denote the operation 'reflection in m' and is written 'as r

_{m}: P → P' or r

_{m}(P) = P'.

**Note:**

**RFLECTION CONSTRUCTION:**

**Reflection of a point in a line:**

**Steps of Construction:**

**REFLECTION OF A LINE SEGMENT IN A LINE**

**Steps of Construction:**

**Reflection of a point in a point:**Let o be the point in which we require to find the reflection of point P.

**Steps of Construction:**

**Reflection of a line segment in a point:**Let O be the point in which we require to find the reflection of line segment PQ.

**Steps of Construction:**

**Live Math Experts: Help with Reflection Assignments - Homework**

**Math Online Tutoring**: Reflection