Prove that the points 0 , 0 , 3 , π 2 and 3 , π 6 form an equilateral triangle.

Let P ≡ 0 , 0 , Q ≡ 3 , π 2 and R ≡ 3 , π 6 be the vertices of ∆ P Q R . On comparing the vertices with r 1 , θ 1 , r 2 , θ 2 and r 3 , θ 3 , we get r 1 as 0 , r 2 as 3 , r 3 as 3 , θ 1 as 0 , θ 2 as π 2 and θ 3 as π 6 . P Q = r 1 2 + r 2 2 - 2 r 1 r 2 cos θ 1 - θ 2 P Q = 0 2 + 3 2 - 2 × 0 × 3 cos π 2 P Q = 3 units Q R = r 2 2 + r 3 2 - 2 r 2 r 3 cos θ 2 - θ 3 Q R = 3 2 + 3 2 - 2 × 3 × 3 cos π 2 - π 6 Q R = 9 + 9 - 18 cos π 3 Q R = 3 units P R = r 1 2 + r 3 2 - 2 r 1 r 3 cos θ 1 - θ 3 P R = 0 2 + 3 2 - 2 × 0 × 3 cos 0 - π 6 P R = 3 units ∵ P Q = Q R = P R ∴ ∆ P Q R is an equilateral triangle.

E M B I B E

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 1 - Coordinates, Lengths of Straight Lines and Areas of Triangles>EXAMPLES II>Q 13

EASY

JEE Advanced

IMPORTANT

Earn 100

Find the area of the quadrilateral the co-ordinates of whose angular points, taken in order, are $(1, 1), (3, 4), (5, - 2)$ and $(4, - 7)$ .

Important Questions on Coordinates, Lengths of Straight Lines and Areas of Triangles

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 1 - Coordinates, Lengths of Straight Lines and Areas of Triangles>EXAMPLES II>Q 14

Find the area of the quadrilateral the co-ordinates of whose angular points, taken in order, are

(- 1, 6), (- 3, - 9), (5, - 8)

and

(3, 9) .

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 1 - Coordinates, Lengths of Straight Lines and Areas of Triangles>EXAMPLES II>Q 15

If $O$ is the origin, and if the co-ordinates of any two points $P_{1}$ and $P_{2}$ are respectively $(x_{1}, y_{1})$ and $(x_{2}, y_{2}),$ prove that
$O P_{1} \cdot O P_{2} \cdot \cos ∠ (P_{1} O P_{2}) = x_{1} x_{2} + y_{1} y_{2} .$