Let's start by identifying the important information given:
Coordinates of point
A are
(1,2).
Equation of the median from
B is
x+y=5.
Equation of the median from
C is
x=4.
The median of a triangle from a vertex passes through the midpoint of the opposite side. Since we are provided with the equations of the medians, we can use this fact to find the coordinates of
B and
C.
First, we find the intersection point of the two medians, which gives the centroid (denoted
G ) of the triangle
â–³ABC. The centroid is the point where all three medians intersect and divides each median in the ratio
2:1.
Given equations of the medians are:
‌x+y=5‌x=4 Substituting
x=4 into the first equation, we get:
4+y=5⟹y=1Thus, the intersection point, which is also the centroid
G, has coordinates
(4,1).
The coordinates of the centroid can also be determined by the formula derived from the section formula:
G=(‌,‌) where
(x1,y1),(x2,y2), and
(x3,y3) are the coordinates of vertices
A,B, and
C respectively.
Since the centroid
G and vertex
A are given, we use the centroid formula:
(‌,‌)=(4,1) Now solving these equations:
‌‌=4⟹5+x2=12⟹x2=7‌‌=1⟹3+y2=3⟹y2=0Thus, the coordinates of vertex
B are
(7,0), but checking the options, it looks like there might have been an error in my calculations or option transcription. Let's check again.
Checking each option's coordinates:
Checking each option's coordinates:
Option A: If coordinates of
B are
(4,1), then the median would not fit other conditior
Option B: For coordinates
(7,−2) :
Midpoint with
C(4,1) should be
(‌,‌)=(‌,−‌)Fits the equation
x+y=5:‌−‌=5✔Hence, the solution is Option
B : Coordinates of
B are
(7,−2).