Question

In the given figure, if the line segment AB is intercepted by the y-axis and x-axis at C and D, respectively, such that AC : AD = 1 : 4 and D is the midpoint of CB. Find the coordinates of D, C and В.

Answer 1

Given: A = (–2, 6). The line AB meets the y-axis at C and the x-axis at D, with AC : AD = 1 : 4 and D the midpoint of CB.

Step-wise calculation:

1. Parametrize C along AD:

`C = A + (1/4) (D - A)` because `(AC)/(AD) = 1/4`

For x-coordinate:

0 = x_C 

= `x_A + (1/4) (x_D - x_A)`

= `-2 + (1/4) (x_D + 2)`

Multiply by 4:

0 = –8 + x_D + 2

⇒ x_D = 6

Hence, D = (6, 0).

2. y-coordinate of C:

`y_C = y_A + (1/4) (y_D - y_A)` 

= `6 + (1/4) (0 - 6)` 

= `6 - 3/2` 

= `9/2`

So, `C = (0, 9/2)`.

3. D is midpoint of C and B.

So, B = 2D – C (coordinate-wise):

x_B = 2 × 6 – 0

= 12

`y_B = 2 xx 0 - 9/2`

= `−9/2`

Thus, `B = (12, -9/2)`.