Question

Find the coordinates of points L, M, and N, the midpoints of the sides ABC.

Answer 1

Given:- ABC is a triangle and L, M and N are the mid-points of the AB, BC, and AC.

To find:- The coordinates of L, M, and N.

Solution:-

To calculate the coordinate of L, M and N. We are going to use the mid-point theorem.

The formula is:

`Mid-point\text{ }coordinate=((x_1+x_2)/(2),(y_1+y_2)/(2))`

(a) First, calculate the coordinate of L.

Here coordinate of A is (0,0) and the coordinate of B is (6q, 6r).

So, the coordinate of L can be calculated as:

`\begin{gathered} Coordinate\text{ }of\text{ }L=((x_1+x_2)/(2),(y_1+y_2)/(2)) \\ Coordinate\text{ }of\text{ }L=((0+6q)/(2),(0+6r)/(2)) \\ Coordinate\text{ }of\text{ }L=((6q)/(2),(6r)/(2)) \\ Coordinate\text{ }of\text{ }L=(3q,3r) \end{gathered}`

(b) Now calculating the coordinate of M.

Here the coordinate of B is (6q, 6r) and the coordinate of C is (6p, 0).

So, the coordinate of M can be calculated as:

`\begin{gathered} Coordinate\text{ }of\text{ }M=((x_1+x_2)/(2),(y_1+y_2)/(2)) \\ Coordinate\text{ }of\text{ }M=((6q+6p)/(2),(6r+0)/(2)) \\ Coordinate\text{ }of\text{ }M=(3q+3p,3r) \end{gathered}`

(c) Now calculating the coordinate of N.

Here the coordinate of A is (0,0) and the coordinate of C is (6p, 0).

So, the coordinate of N can be calculated as:

`\begin{gathered} Coordinate\text{ }of\text{ }N=((x_1+x_2)/(2),(y_1+y_2)/(2)) \\ Coordinate\text{ }of\text{ }N=((0+6p)/(2),(0+0)/(2)) \\ Coordinate\text{ }of\text{ }N=(3p,0) \end{gathered}`

Final answer:-

Therefore, the answer is:

`\begin{gathered} Coordinate\text{ }of\text{ }L=(3q,3r) \\ Coordinate\text{ }of\text{ }M=(3q+3p,3r) \\ Coordinate\text{ }of\text{ }N=(3p,0) \end{gathered}`