Question

The midpoint of side BC of triangle ABC, with A (1, -4) and the mid-points of the sides through A being (2, -1) and (0, -12) is

Answer 1

Hint: In the above question we have a triangle ABC in which D, E, and F are midpoints. Side AB has a mid-point D, Side AC has a midpoint E, and Side BC has a midpoint F. Co-ordinate of A, D, and E is given. We will use the midpoint formula to solve this question.

Complete step-by-step answer:
Drawing figure from the given data:
     

From the figure, we know that;
D is the midpoint of AB.
E is the midpoint of AC.
F is the midpoint of BC.
In triangle ABC,
Coordinates of A (1, -4)
Coordinates of D (2, -1)
Coordinates of E (0, -12)
We need to find the Coordinates of F which is a midpoint of BC.
Let the Coordinates of B be $\left(x_2,y_2\right)$, C be $\left(x_3,y_3\right)$, and F be $\left(x,y\right)$.
Now, using midpoint formula, we will try to find the coordinates of B
Coordinates of B $={\displaystyle \frac{1+x_2}{2}}=2$ and ${\displaystyle \frac{-4+y_2}{2}}=-1$
After solving above equation, we get;
Coordinates of B (3, 2)
Now, using midpoint formula, we will try to find the coordinates of C
Coordinates of C $={\displaystyle \frac{1+x_3}{2}}=0$ and ${\displaystyle \frac{-4+y_3}{2}}=-12$
After solving above equation, we get;
Coordinates of C (-1, -20)
Now, using midpoint formula, we will try to find the coordinates of F
Coordinates of F $={\displaystyle \frac{x_2+x_3}{2}}=x$ and ${\displaystyle \frac{y_2+y_3}{2}}=y$
Putting the values in the above equations, we get;
$={\displaystyle \frac{3-1}{2}}=x$ and ${\displaystyle \frac{2-20}{2}}=y$
After solving above equation, we get;
Coordinates of F (1, -5)

Note: In this type of problems usually students make mistakes by considering the end point coordinates as equals to the sum of another endpoint and midpoint coordinates. The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.