Question

the midpoints of BC, CA, AB of a triangle ABC are of D(2, 1), E(-1, -3) and F(4, 5) respectively. Find the coordinates of A, B and C

Answer 1

Answer:

Step-by-step explanation:

Let the coordinates of vertices A, B and C of the ∆ABC be (x1, y1), (x2, y2) and (x3, y3) respectively. Let D (3, 4), E (4, 1) and F (2, 0) be the mid parts of sides BC, AC and AB respectively

BY USING SECTION FORMULA,

Mid points always divide in in 1:1.

So,

for BC

$\frac{x2+x3}{2}=2$ ,$\frac{y2+y3}{2}=1$

FOR CA

$\frac{x3+x1}{2}=-1 , \frac{y3+y1}{2} =3$

FOR AB

$\frac {x1+x2}{2}=4 ,,\frac{y1+y2}{2}=5$

⇒x2+x3 =4  ...............(1)

  y2+y3=2   ...............(2)

  x3+x1=-2    ...............(3)

  y3+y1=6     ...............(4)

  x1+x2=8     ...............(5)

  y1+y2=10    ..............(6)

Adding equations (1), (3) and (5)

2 (x1 + x2 + x3) = 10

⇒ x1 + x2 + x3 = 5     … (7)

Subtracting equation (1), (3) and (5) from equation (7), it is obtained

x1 = 5– 4 = 1

x2 = 5 – (-2) = 7

x3 = 5 – 8 = -3

Adding equations (2), (4) and (6)

2 (y1 + y2 + y3) = 18

⇒ y1 + y2 + y3 =9    … (8)

Subtracting equation (2), (4) and (6) from equation (8), it is obtained

y1 = 9– 2 = 7

y2 = 9 – 6 = 3

y3 = 9 – 10 = -1

Thus, the vertices of the triangles are (1, 7), (7, 3) and (-3, -1).

PRACTICE YOURSELF ON COPY AND BY MAKING DIAGRAM TO CLEAR CONCEPTS.

THIS QUESTION IS FREQUENTLY ASKED IN BOARD EXAM.