The midpoints of the sides of triangle ABC are the points D, E, and F. Given that Class 9

The midpoints of the sides of triangle ABC are the points D, E, and F. Given that Class 9

Question 1.

The midpoints of the sides of triangle ABC are the points D, E, and F. Given that the coordinates of D, E, and F are (5, 1), (6, 5), and (0, 3), respectively, find the coordinates of A, B and C.

Solution:

Let A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃)

Now,

5 = [latex]\frac{x_1+x_2}{2}[/latex] and 1 = [latex]\frac{y_1+y_2}{2}[/latex]

∴ x₁ + x₂ = 10 and y₁ + y₂ = 2 ....(i)

Similarly,

x₁ + x₃ = 0 and y₁ + y₃ = 6 and

x₂ + x₃ = 12 and y₂ + y₃ = 10

Subtracting the above equations, we get

x₁ - x₂ = -12 and y₁ - y₂ = -4 ... (ii)

Adding (i) and (ii), we get

2x₁ = -2 and 2y₁ = -2

∴ x₁ = -1 and y₁ = -1

From (i), we get

-1 + x₂ = 10 and -1 + y₂ = 2

∴ x₂ = 11 and y₂ = 3

Now, x₂ + x₃ = 12 and y₂ + y₃ = 10

∴ 11 + x₃ = 12 and 3 + y₃ = 10

∴ x₃ = 1 and y₃ = 7

∴ The coordinates of A, B and C are (-1, -1), (11, 3) and (1, 7) respectively.