Shortcut for Finding the Equation of a Line as a Median of a Triangle

Question

For a National Board Exam:

The points A(1,0), B(9,2), C(3,6) are vertices of a triangle. Which of the following is an equation of one of the medians?

Choices are:

A. ${7x-y=23}$

B. ${x-7y=23}$

C. ${7x + y = 23}$

D. ${x+7y=23}$

Answer is D. x+7y=23

Ok. I know how to solve this manually. Get midpoint and slope for each pair and simply put them into the point slope form... but for my exam i really need to solve this really fast cuz I have another 99 items to go in only 2 hours... Is there a way that I can solve this in a more clever and faster manner?

Answer 1

Medians intersect in centroid. That is easy to find, average of x and y co ordinates. Easy to inspect that only one of the lines passes. Of course, in a different question, of the given choices, if more than 1 line passes, then check out the vertices.

Answer 2

Here's a way to save a bit of time: the three midpoints are $(5,1)$, $(2,3)$, and $(6,4)$. It's easy to check that the only one of the answer choices that passes through any of these points is D.

Answer 3

Since, a median is a line joining any of the vertices $A(1, 0)$, $B(9, 2)$ & $C(3, 6)$ of triangle to the mid-point of its opposite side hence equation of the median should be satisfied by a vertex.

In the given question here, $x+7y=23$ is satisfied by the vertex $B(9, 2)$ hence it is a median passing through the vertex $(7, 2)$