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Points a,b, and c make the triangle ABC and are at the coordinates A(-2,9), B(-33,13) and c(-21,25) point D is the midpoint of BC and AD is a median of ABC, the equation of the median can be given by ax+by=c where A B and C

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Answer:


2x+5y=41

Explanation:

Median of a triangle: A line segment that connects a vertex of a triangle to the midpoint of the opposite side.

Vertex: The point where any two sides of a triangle meet.

Given vertices of a triangle:

  • A = (-2, 9)
  • B = (-33, 13)
  • C = (-21, 25)

Step 1

Find the midpoint of BC (Point D) by using the Midpoint formula.

Midpoint between two points


\textsf{Midpoint}=\left((x_2+x_1)/(2),(y_2+y_1)/(2)\right)\quad \textsf{where}\:(x_1,y_1)\:\textsf{and}\:(x_2,y_2)\:\textsf{are the endpoints}}\right)

Define the endpoints:


  • \text{Let }(x_1,y_1)=\sf B=(-33,13)

  • \text{Let }(x_2,y_2)=\sf C=(-21,25)

Substitute the defined endpoints into the formula:


\textsf{Midpoint of BC}=\left((-21-33)/(2),(25+13)/(2)\right)=(-27,19)

Therefore, D = (-27, 19).

Step 2

Find the slope of the median (line AD) using the Slope formula.

Define the points:


  • \textsf{let}\:(x_1,y_1)=\sf A=(-2,9)

  • \textsf{let}\:(x_2,y_2)=\sf D=(-27,19)

Substitute the defined points into the Slope formula:


\implies \textsf{slope}\:(m)=(y_2-y_1)/(x_2-x_1)=(19-9)/(-27-(-2))=-(2)/(5)

Therefore, the slope of the median is -²/₅.

Step 3

Substitute the found slope and one of the points into the Point-slope formula to create an equation for the median.


\implies y-y_1=m(x-x_1)


\implies y-9=-(2)/(5)(x-(-2))

Simplify and rearrange the equation so it is in standard form Ax+By=C:


\implies 5(y-9)=-2(x+2)


\implies 5y-45=-2x-4


\implies 2x+5y-45=-4


\implies 2x+5y=41

Conclusion

Therefore, the equation of the median is:

2x + 5y = 41

Points a,b, and c make the triangle ABC and are at the coordinates A(-2,9), B(-33,13) and-example-1
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