Definitions [4]
The two mutually perpendicular number lines intersecting each other at their zeroes are called rectangular axes or coordinate axes, or axes of reference.
The position of a point in a plane is expressed by a pair of numbers, one concerning the x-axis and the other concerning the y-axis. called co-ordinates.
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x → distance from y-axis (abscissa)
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y → distance from x-axis (ordinate)
Reflection is a transformation in which the image of a point is formed at the same distance on the opposite side of a line (mirror).
An invariant point is a point whose coordinates do not change after a transformation.
Formulae [1]
| Sr. No. | Name | Condition | Formula |
|---|---|---|---|
| i. | Distance Formula | Two points P(x₁, y₁), Q(x₂, y₂) | \[\mathrm{d(PQ)}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\] |
| a. | Internal Division | P divides AB in the ratio m: n | \[\mathrm{P\equiv\left(\frac{mx_{2}+nx_{1}}{m+n},\frac{my_{2}+ny_{1}}{m+n}\right)}\] |
| b. | Midpoint Formula | P is the midpoint of AB | \[\mathrm{P}\equiv\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)\] |
| c. | External Division | P divides AB externally in m: n | \[\mathrm{P\equiv\left(\frac{mx_{2}-nx_{1}}{m-n},\frac{my_{2}-ny_{1}}{m-n}\right)}\] |
| iii. | Centroid Formula | Triangle with vertices A, B, C | \[\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)\] |
Key Points
Sign Convention
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Right of y-axis → +x
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Left of y-axis → −x
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Above x-axis → +y
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Below x-axis → −y
Standard Line Results
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x = 0 → y-axis
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y = 0 → x-axis
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x = a → line parallel to the y-axis
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y = b → line parallel to the x-axis
Quadrant Reminder
| Quadrant | Sign of (x, y) |
|---|---|
| I | (+, +) |
| II | (−, +) |
| III | (−, −) |
| IV | (+, −) |
In x-axis (y = 0)
(x,y) → (x,−y)
In y-axis (x = 0)
(x, y) → (−x, y)
In origin
(x, y)→(−x,−y)
Reflection in Parallel Lines:
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In line y = a
(x, y)→(x, 2a − y) -
In line x = a
(x, y)→(2a − x, y)
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Reflection in the x-axis
Points on x-axis (x,0) -
Reflection in the y-axis
Points on y-axis (0,y) -
Reflection in origin
Only the origin (0,0) -
Reflection in line y = a
Points lying on the line y = a -
Reflection in line x = a
Points lying on the line x = a
| Combination | Result |
|---|---|
| (Rx Ry) | (Ro) |
| (Ry Rx) | (Ro) |
| (Rx Ro) | (Ry) |
| (Ry Ro) | (Rx) |
Important Questions [2]
- Use graph sheet to Solution this question. Take 2 cm = 1 unit alogn both the axes. Plot A, B, C where A(0, 4), B(1, 1) and C(4, 0) Reflect A and B on the x-axis and name them as E and D respectively
- Use graph sheet for this question. Take 2 cm = 1 unit along the axes. Plot A(0, 3), B(2, 1) and C(4, –1). Reflect point B and C in y-axis and name their images as B' and C' respectively.
