# ABOUT COORDINATE GEOMETRY. PLZ EXPLN IT IN DETAIL.......IN TRIANGLE ABC, COORDINATES OF A ARE (1,2) & THE EQUATIONS OF MEDIANS THROUGH B & C ARE x + y=5 & x = 4 RESPECTIVELY, SO...

ABOUT COORDINATE GEOMETRY. PLZ EXPLN IT IN DETAIL.......

IN TRIANGLE ABC, COORDINATES OF A ARE (1,2) & THE EQUATIONS OF MEDIANS THROUGH B & C ARE x + y=5 & x = 4 RESPECTIVELY, SO THE COORDINATES OF B & C ARE

A) (0,0) (2,0) B) (7,-2) (4,3) C) (4,5) (-2,-1) D) NONE

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### 1 Answer

*IN TRIANGLE ABC, COORDINATES OF A ARE (1,2) & THE EQUATIONS OF MEDIANS THROUGH B & C ARE x + y=5 & x = 4 RESPECTIVELY, SO THE COORDINATES OF B & C ARE*

*A) (0,0) (2,0) B) (7,-2) (4,3) C) (4,5) (-2,-1) D) NONE*

Let M be the midpoint of AC, and N the midpoint of AB. Also remember that a median of a triangle is drawn through a vertex to the midpoint of the side opposite that vertex.

Now let the coordinates of C be `(x_1,y_1)` . Since the median through C is the line x=4, the x-coordinate of C is 4. Using the midpoint formula, we find that M=`((4+1)/2,(y_1+2)/2)` or `(5/2,(y_1+2)/2)` . Now M also lies on the line x+y=5, so x=5/2 implies the y-coordinate of M is also 5/2, thus `(y_1+2)/2=5/2` so `y_1=3` and **the coordinates of C are (4,3).**

Now we consider the point B=`(x_2,y_2)` . Again applying the midpoint formula we have `N=((x_2+1)/2,(y_2+2)/2)` . But the x-coordinate for N is 4, so `(x_2+1)/2=4 => x_2=7` . Since B lies on the line x+y=5, the y-coordinate is -2, so **the coordinates for B are (7,-2).**

**So the solution is B) (7,-2),(4,3).**

** Since this was given in multiple choice format, you could have checked each potential solution. A quick sketch shows B is the only reasonable answer (Also the x-coordinate for C must be 4 by the definition of a median), and you can check that the midpoint of the segment from (1,2) to (4,3) actually lies on the line x+y=5, and that the midpoint of the segment from (1,2) to (7,-2) lies on the line x=4. **