Question

Find the area of the triangle PQR with Q (3, 2) and the mid-points of the sides through Q being (2, -1) and (1, 2).

Answer 1

The area of the triangle PQR is = 12 square units

Let P be (x,y)

Q is (3,2)

Let the mid point of PQ be (2,-1)

∴ (x+3)/2 = 2

⇒ x+3 = 4

⇒ x = 1

And, (y+2)/2 = -1

⇒ y+2 = -2

⇒ y = -4

So, P is (1,-4)

Let R be (a,b)

Let the mid point of QR be (1,2)

∴ (a+3)/2 = 1

⇒ a+3 = 2

⇒ a = -1

And, (b+2)/2 = 2

⇒ b+2 = 4

⇒ b = 2

So, R is (-1,2)

Now, we have to find out the area of the triangle PQR where P(1,-4) , Q(3,2), R(-1,2)

Area of the triangle is given by = (1/2) [x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)]

where,

x₁ = 1

x₂ = 3

x₃ = -1

y₁ = -4

y₂ = 2

y₃ = 2

Replacing the values, we get

Area = (1/2) [1(2-2) + 3(2+4) -1(-4-2)]

=  (1/2) [24]

= 12 square units