Find the area of the hexagon whose consecutive vertices are 5 , 0 , 4 , 2 , 1 , 3 , - 2 , 2 , - 3 , - 1 and 0 , - 4 .

We have, x 1 = 5 , y 1 = 0 ; x 2 = 4 , y 2 = 2 ; x 3 = 1 , y 3 = 3 ; x 4 = - 2 , y 4 = 2 ; x 5 = - 3 , y 5 = - 1 and x 6 = 0 , y 6 = - 4 . We know that Area of hexagon = 1 2 x 1 y 1 x 2 y 2 + x 2 y 2 x 3 y 3 + x 3 y 3 x 4 y 4 + x 4 y 4 x 5 y 5 + x 5 y 5 x 6 y 6 + x 6 y 6 x 1 y 1 = 1 2 5 0 4 2 + 4 2 1 3 + 1 3 - 2 2 + - 2 2 - 3 - 1 + - 3 - 1 0 - 4 + 0 - 4 5 0 On expanding, we get = 1 2 10 - 0 + 12 - 2 + 2 + 6 + 2 + 6 + 12 - 0 + 0 + 20 = 1 2 68 = 34 square units

Three vertices of a triangle are A ( 1 , 2 ) , B ( - 3 , 6 ) and C ( 5 , 4 ) . If D , E and F are the mid-points of the sides opposite to the vertices A , B and C , respectively, show that the area of triangle A B C is four times the area of triangle D E F .

Given: vertices of triangle A B C are A 1 , 2 , B - 3 , 6 and C 5 , 4 . ∵ D is the midpoint of B C . ∴ Coordinates of point D = - 3 + 5 2 , 6 + 4 2 = 1 , 5 ∵ E is the midpoint of A C ∴ Coordinates of point E = 1 + 5 2 , 2 + 4 2 = 3 , 3 ∵ F is the midpoint of A B ∴ Coordinates of the point F = 1 - 3 2 , 2 + 6 2 = - 1 , 4 As we know if x 1 , y 1 , x 2 , y 2 and x 3 , y 3 are the coordinates of the vertices of the triangle, then the area of the triangle is given by 1 2 x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 . Now, area of the triangle A B C = 1 2 1 2 1 - 3 6 1 5 4 1 = 1 2 1 6 - 4 - 2 - 3 - 5 + 1 - 12 - 30 = 1 2 2 + 16 - 42 = 12 square units Now, area of triangle D E F = 1 2 1 5 1 3 3 1 - 1 4 1 = 1 2 1 3 - 4 - 5 3 + 1 + 1 12 + 3 = 1 2 - 1 - 20 + 15 = 3 square units. ∴ Area of the triangle A B C is four times the area of the triangle D E F .

If three points x 1 , y 1 , x 2 , y 2 and x 3 , y 3 lie on the same line, prove that y 2 - y 3 x 2 x 3 + y 3 - y 1 x 3 x 1 + y 1 - y 2 x 1 x 2 = 0 .

Given three points x 1 , y 1 , x 2 , y 2 and x 3 , y 3 lie on the same line. Then, area of Δ = 0 ⇒ 1 2 x 1 y 2 - y 3 + x 2 y 3 - y 1 + x 3 y 1 - y 2 = 0 ⇒ x 1 y 2 - y 3 + x 2 y 3 - y 1 + x 3 y 1 - y 2 = 0 Now dividing on both sides with ' x 1 x 2 x 3 ', we get, ⇒ x 1 y 2 - y 3 x 1 x 2 x 3 + x 2 y 3 - y 1 x 1 x 2 x 3 + x 3 y 1 - y 2 x 1 x 2 x 3 = 0 x 1 x 2 x 3 ∴ y 2 - y 3 x 2 x 3 + y 3 - y 1 x 1 x 3 + y 1 - y 2 x 1 x 2 = 0 Hence proved.

E M B I B E

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 1 - Cartesian System of Rectangular Coordinates>1.4>Q 3

MEDIUM

JEE Main

IMPORTANT

Earn 100

Find the area of the pentagon whose vertices are $(4, 3), (- 5, 6), (0, - 7), (3, - 6),$ $(- 7, - 2)$

Important Questions on Cartesian System of Rectangular Coordinates

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 1 - Cartesian System of Rectangular Coordinates>1.4>Q 4

Find the area of the hexagon whose consecutive vertices are

= \frac{1}{2} |20 + 70|

and

(0, - 4) .

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 1 - Cartesian System of Rectangular Coordinates>1.4>Q 6

Three vertices of a triangle are

= \frac{1}{2} |4 (- 1) + 0 (- 9) + 3 (10)|

and

C (5, 4) .

If

D, E

and

F

are the mid-points of the sides opposite to the vertices

A, B

and

= 13

respectively, show that the area of triangle

A B C

is four times the area of triangle

D E F

.

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 1 - Cartesian System of Rectangular Coordinates>1.4>Q 7

Find the area of a triangle $ABC$ if the coordinates of the middle points of the sides of the triangle are $(- 1, - 2), (6, 1)$ and $(3, 5) .$

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 1 - Cartesian System of Rectangular Coordinates>1.4>Q 8

The vertices of $Δ A B C$ are $A (3, 0), B (0, 6)$ and $(6, 9) .$ A straight line $D E$ divides $A B$ and $A C$ in the ratio $1 : 2$ at $D$ and $E$ respectively, prove that $\frac{Δ A B C}{Δ A D E} = 9$

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 1 - Cartesian System of Rectangular Coordinates>1.4>Q 9

If

(t, t - 2), (t + 3, t)

and

(t + 2, t + 2)

are the vertices of a triangle, show that its area is independent of

t .

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 1 - Cartesian System of Rectangular Coordinates>1.4>Q 10

If

A (x, y), B (1, 2)

and

C (2, 1)

are the vertices of a triangle of area

6

square units, show that

x + y = 15

or

- 9

.

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 1 - Cartesian System of Rectangular Coordinates>1.4>Q 11

Prove that the points

(a, b + c), (b, c + a)

and

(c, a + b)

are collinear.

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 1 - Cartesian System of Rectangular Coordinates>1.4>Q 12

If three points $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ lie on the same line, prove that $\frac{y_{2} - y_{3}}{x_{2} x_{3}} + \frac{y_{3} - y_{1}}{x_{3} x_{1}} + \frac{y_{1} - y_{2}}{x_{1} x_{2}} = 0$ .