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**The X â€“ Y Plane**

The number lines, when drawn as shown in X â€“ Y plane below, are called "**axes**". The horizontal number line

is called the "**x-axis**", the vertical one is the **y-axis.**

**Abscissa** is the xâ€“coordinate of a point can be defined either as its distance along the xâ€“axis, or as its

perpendicular distance from the yâ€“axis.

**Ordinate** is the y-coordinate of a point can be defined either as its distance along the y-axis, or as its

perpendicular distance from the x-axis.

In the figure given below, OX and OY are two straight lines which are perpendicular to each other and which

intersect at the point O. OX is known as the xâ€“axis, and OY is known as the yâ€“axis. You can see that the two

axes divide the plane into four regions as above. The four regions are known as **Quadrants **and are named I

Quadrant, II Quadrant, III Quadrant and IV Quadrant as shown.

(A) Co-ordinate of the origin is (0, 0).

(B) Any point on the x axis can be taken as (a, 0)

(C) Any point on the y axis can be taken as (0, b)

**Ex.1 In which quadrant is (x, y), such that x y < 0?**

**Sol. **The points (x, y), with xy < 0 means the product of abscissa and ordinate should be negative which can

be possible only when one is positive and other is negative, such as (â€“ 2, 5) or (4, â€“ 6) and this will lie

in the Quadrants II and IV.

**(i) Distance formula:****Ex.2 A (a, 0) and B (3a, 0) are the vertices of an equilateral triangle ABC. What are the coordinates of C? (1) (a, aâˆš3) **

**(2) (aâˆš3, 2a) **

**(3) (aâˆš3, 0) **

**(****4) (2a, + aâˆš3) **

**(5) None of these**

**Sol. **

Now, the vertex C will be such that AC = BC = 2a

âˆ´ if (x, y) are the co-ordinates of C

or x = 2a.

**(ii) Section formula:**

The point which divides the join of two distinct points A (x_{1}, y_{1})

and B (x_{2}, y_{2}) in the ratio m_{1} : m_{2} Internally, has the co-ordinates

In particular, the mid-point of the segment joining A (x_{1}, y_{1}) and B (x_{2}, y_{2}) has the co-ordinates

**Ex.3 Find the points A and B which divide the join of points (1, 3) and (2, 7) in ratio 3 : 4 both internally & externally respectively .**

**Sol.**

**(iii) Centroid and Incentre formulae:**

**Centroid: **It is the point of intersection of the medians of a triangle.

**Incentre:** It is the point of intersection of the internal angle bisectors of the angles of a triangle.

If A be the vertices of a triangle, then its **centroid** is given by

and the incentre by,

Where a = | BC |, b = | CA | and c = | AB |.

**Ex.4 If (2, 3), (3, a), (b, â€“ 2) are the vertices of the triangle whose centroid is (0, 0), then find the value of a and b respectively. (1) â€“ 1, â€“ 4 **

**(2) â€“ 2, â€“ 5 **

**(3) â€“ 1, â€“ 6 **

**(4) â€“ 1, â€“ 5 **

**(5) None of these**

**Sol. **x co-ordinate of the centroid =

y co-ordinate of the centroid =

a = â€“ 1 **Answer: (4)**

**(iv) Area of triangle:**

**Ex.5 A triangle has vertices A (2, 2), B (5, 2) and C (5, 6). What type of triangle it is ?**

**Sol.** By the distance formula

According to Pythagorean if the sum of the square of two sides are equal to the square of the third side

then the triangle is a right-angled triangle.

d(AC)^{2} = d(AB)^{2} + d(BC)^{2}

(5)2 = (3)^{2} + (4)^{2}

Therefore, the triangle is right-angled.**Important**

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