Math Problem Statement

Question 4. Triangle ABC has vertices A(10, 2), B(4, −2), and C(−4, 6). Determine the equation of the median from the vertex A.

Solution

To determine the equation of the median from vertex AA of triangle ABCABC, follow these steps:

Step 1: Find the midpoint of side BCBC.

The median from vertex AA will pass through the midpoint of the opposite side BCBC. The coordinates of the midpoint MM of a line segment with endpoints B(x1,y1)B(x_1, y_1) and C(x2,y2)C(x_2, y_2) can be found using the midpoint formula:

M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

For points B(4,2)B(4, -2) and C(4,6)C(-4, 6), the coordinates of the midpoint MM are:

M=(4+(4)2,2+62)=(02,42)=(0,2)M = \left(\frac{4 + (-4)}{2}, \frac{-2 + 6}{2}\right) = \left(\frac{0}{2}, \frac{4}{2}\right) = (0, 2)

So, the midpoint MM of side BCBC is (0,2)(0, 2).

Step 2: Determine the equation of the line passing through A(10,2)A(10, 2) and M(0,2)M(0, 2).

The equation of a line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) can be written in the point-slope form:

yy1=m(xx1)y - y_1 = m(x - x_1)

where mm is the slope of the line, given by:

m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}

For the points A(10,2)A(10, 2) and M(0,2)M(0, 2):

m=22010=010=0m = \frac{2 - 2}{0 - 10} = \frac{0}{-10} = 0

Since the slope m=0m = 0, the line is horizontal. The equation of a horizontal line through the point (10,2)(10, 2) (or any point on the line) is simply:

y=2y = 2

Final Answer:

The equation of the median from vertex AA of triangle ABCABC is:

y=2y = 2

Would you like more details or have any questions? Here are some related questions you might consider:

  1. How do you find the equation of a median when the vertices have different y-coordinates?
  2. What is the significance of the median in a triangle?
  3. How can you find the centroid of a triangle using the medians?
  4. How would the equation of the median change if point AA had different coordinates?
  5. How do you find the equation of a median in 3D space?

Tip: When working with triangles, always double-check your midpoint calculations, as this can affect the equation of the median.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Geometry
Midpoint Formula
Equation of a Line

Formulas

Midpoint formula
Point-slope form of a line equation

Theorems

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Suitable Grade Level

Grades 9-12