A day full of math games & activities. Find one near you.

The mid-points D, E, F of the sides of a triangle ABC are (3, 4), (8, 9) and (6, 7). Find the coordinates of the vertices of the triangle

Solution:

Given, the midpoints of sides of ∆ ABC are D(3, 4) E(8, 9) and F(6, 7)

We have to find the coordinates of the vertices of the ∆ ABC.

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

the-mid-points-d-e-f-of-the-sides-of-a-triangle-abc-are-3-4-8-9-and-6-7-find-the-coordinates-of-the-vertices-of-the-triangle

The coordinates of the mid-point of the line segment joining the points P (x₁ , y₁) and Q (x₂ , y₂) are [(x₁ + x₂)/2, (y₁ + y₂)/2]

Midpoint of AB = D

[(x₁ + x₂)/2, (y₁ + y₂)/2] = (3, 4)

Now, (x₁ + x₂)/2 = 3

x₁ + x₂ = 6 ----------- (1)

Also, (y₁ + y₂)/2 = 4

y₁ + y₂ = 8 ------------ (2)

Midpoint of BC = E

[(x₂ + x₃)/2, (y₂ + y₃)/2] = (8, 9)

Now, (x₂ + x₃)/2 = 8

x₂ + x₃ = 16 ------------------ (3)

Also, (y₂ + y₃)/2 = 9

y₂ + y₃ = 18 -------------------- (4)

Midpoint of AC = F

[(x₁ + x₃)/2, (y₁ + y₃)/2] = (6, 7)

Now, (x₁+x₃)/2 = 6

x₁ + x₃ = 12 ---------------- (5)

Also, (y₁ + y₃)/2 = 7

y₁ + y₃ = 14 ---------------- (6)

Adding (1), (3) and (5) we get,

2(x₁ + x₂ + x₃) = 6 + 16 + 12

2(x₁ + x₂ + x₃) = 34

x₁ + x₂ + x₃ = 17 --------------- (7)

Substitute (1) in (7),

6 + x₃ = 17

x₃ = 11

Substitute (2) in (7),

x₁ + 16 = 17

x₁ = 17 - 16

x₁ = 1

Substitute (3) in (7),

x₂ + 12 = 17

x₂ = 17 - 12

x₂ = 5

Adding (2), (4) and (6) we get,

2(y₁ + y₂ + y₃) = 8 + 18 + 14

2(y₁ + y₂ + y₃) = 40

y₁ + y₂ + y₃ = 20 ------------------ (8)

Substitute (2) in (8),

8 + y₃ = 20

y₃ = 20 - 8

y₃ = 12

Substitute (4) in (8),

y₁ + 18 = 20

y₁ = 20 - 18

y₁ = 2

Substitute (6) in (8),

y₂ + 14 = 20

y₂ = 20 - 14

y₂ = 6

Therefore, the vertices of ∆ ABC are A(1, 2) B(5, 6) and C(11, 12).

✦ Try This: The mid-points of the sides of a triangle are (4, 5), (6, 7) and (3, 4). Find the coordinates of the vertices of the triangle.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7

NCERT Exemplar Class 10 Maths Exercise 7.4 Sample Problem 1

The mid-points D, E, F of the sides of a triangle ABC are (3, 4), (8, 9) and (6, 7). Find the coordinates of the vertices of the triangle

Summary:

The mid-points D, E, F of the sides of a triangle ABC are (3, 4), (8, 9) and (6, 7). The coordinates of the vertices of the triangle are A(1, 2) B(5, 6) and C(11, 12)

☛ Related Questions: