Question

The midpoints of the sides BC, CA and AB of a ΔАBC are D(3, 4), E(8, 9) and F(6, 7) respectively. Find the coordinates of the vertices of the triangle.

Answer 1

Given: Midpoints: D(3, 4) is midpoint of BC; E(8, 9) is midpoint of CA; F(6, 7) is midpoint of AB.

Step-wise calculation:

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

2. Midpoint equations:

D: `(x_2 + x_3)/2 = 3, (y_2 + y_3)/2 = 4` 

⇒ x₂ + x₃ = 6, y₂ + y₃ = 8

E: `(x_3 + x_1)/2 = 8, (y_3 + y_1)/2 = 9` 

⇒ x₃ + x₁ = 16, y₃ + y₁ = 18

F: `(x_1 + x_2)/2 = 6, (y_1 + y_2)/2 = 7` 

⇒ x₁ + x₂ = 12, y₁ + y₂ = 14

3. Add the three x-equations:

(x₂ + x₃) + (x₃ + x₁) + (x₁ + x₂) = 6 + 16 + 12 = 34 

⇒ 2(x₁ + x₂ + x₃) = 34 

⇒ x₁ + x₂ + x₃ = 17

x₁ = 17 – (x₂ + x₃) 

= 17 – 6

= 11

x₂ = 17 – (x₃ + x₁) 

= 17 – 16

= 1

x₃ = 17 – (x₁ + x₂) 

= 17 – 12

= 5

4. Add the three y-equations:

8 + 18 + 14 = 40 

⇒ 2(y₁ + y₂ + y₃) = 40 

⇒ y₁ + y₂ + y₃ = 20

y₁ = 20 – (y₂ + y₃) 

= 20 – 8

= 12

y₂ = 20 – (y₃ + y₁) 

= 20 – 18

= 2

y₃ = 20 – (y₁ + y₂) 

= 20 – 14

= 6

5. Check midpoints:

BC midpoint = `((1 + 5)/2, (2 + 6)/2) = (3, 4)`

CA midpoint = `((5 + 11)/2, (6 + 12)/2) = (8, 9)`

AB midpoint = `((11 + 1)/2, (12 + 2)/2) = (6, 7)`

The vertices of ΔABC are A(11, 12), B(1, 2), C(5, 6).