Class 10 Maths notes for Coordinate geometry

1) Find the distance of point (3,4) and ( 4,3) from Origin?
Solution
Distance from Origin is given by
$D=\sqrt {x^2 + y^2}$
for (3,4),
$D=\sqrt {3^2 + 4^2} =5$
for (4,3),
$D=\sqrt {4^2 + 3^2} =5$

Practice Questions
1) Arrange these points in ascending order of distance from origin
a) (1,2)
b) (5,5)
c) (-1, 3)
d) (1/2, 1)
e) (.5 ,.5)
f) (-1,1)

2) Find the distance between the points
a) (1,2) and (3,4)
b) (5,6) and ( 1,1)
c) (9,3) and ( 3,1)

Solved Examples

Question 1
The points (4, 5), (7, 6) and (6, 3) are collinear. True or false
Solution
False. Since the area of the triangle formed by the points is 4 sq. units, the points are not collinear

Question 2
If the mid-point of the line segment joining the points A (3, 4) and B (k, 6) is P (x, y) and
$x + y - 10 = 0$, find the value of k.
Solution
Mid-point of the line segment joining A (3, 4) and B (k, 6) = $\frac {(3+k)}{2}$, 5
Now mid -point is P(x,y)
$x=\frac {(3+k)}{2}$ or 3+k=2x
y=5
Since $x + y - 10 = 0$, we have
$\frac {3+k}{2} +5 - 10 = 0$
3 + k = 10
or k=7

Question 3
ABCD is a parallelogram with vertices A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃). Find the coordinates of the fourth vertex D in terms of x₁,x₂,x₃,y₁,y₂,y₃
Solution
Let the coordinates of D be (x, y). We know that diagonals of a parallelogram bisect each other.
Therefore, mid-point of AC = mid-point of BD
$[ \frac{(x_1 + x_3)}{2} ,\frac{(y_1 + y_3)}{2}]=[ \frac{(x + x_2)}{2} ,\frac {(y + y_2)}{2}]$
i.e $x_1 + x_3 = x_2 + x$ and $y_1 + y_3 = y_2 + y$
$x= x_1 + x_3 - x_2$ and $y=y_1+ y_3 -y_2$
So coordinates are $(x_1 + x_3 - x_2, y_1+ y_3 - y_2)$

Question 4<
The area of a triangle with vertices (a, b + c), (b, c + a) and (c, a + b) is
Solution


$A=\frac {1}{2} [a(c+a -a-b) + b(a+b-b-c) + c( b+c -c-a)]$
$A=\frac {1}{2} [ac-bc + ba-bc + bc-ac)]$
A=0

Crossword Puzzle

Across
1. the distance of point(3,4) from origin
3. x-coordinate is called
5. the y -axis is a .....line
6. The (-6,-2) lies in .....quadrant
7. the x-axis is a ......line
9. The point of intersection of axis
10. The figure formed by joining the point (0,0), (2,0),(2,2) and (0,2)
Down
2. the coordinates (3,0) (-3,0) and $(0,3 \sqrt {3})$ formed an ...... triangle
4. The three point are said to be ....if the area of triangle formed by them is zero
8. y-coordinate is called
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