Find The Equation Of The Median Of The Triangle Formed By The Points $(2,2),(2,8)$, And $(-6,2)$, Drawn From The First Vertex \$(2,2)$[/tex\].Answer: $3x + 4y - 14 = 0$

Introduction

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. The median divides the triangle into two equal areas and is also the line of symmetry of the triangle. In this article, we will find the equation of the median of a triangle formed by three given points.

The Problem

We are given three points: $(2,2)$, $(2,8)$, and $(-6,2)$. We need to find the equation of the median of the triangle formed by these points, drawn from the first vertex $(2,2)$.

Step 1: Find the Midpoint of the Opposite Side

To find the equation of the median, we first need to find the midpoint of the opposite side. The opposite side is the line segment joining the points $(2,8)$ and $(-6,2)$. The midpoint of this line segment can be found using the midpoint formula:

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

where $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of the two points.

Plugging in the values, we get:

$$\left(\frac{2+(-6)}{2},\frac{8+2}{2}\right)$$

$$=\left(\frac{-4}{2},\frac{10}{2}\right)$$

$$=(-2,5)$$

Step 2: Find the Slope of the Median

The slope of the median can be found using the slope formula:

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

where $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of the two points.

In this case, the two points are $(2,2)$ and $(-2,5)$. Plugging in the values, we get:

$$m=\frac{5-2}{-2-2}$$

$$=\frac{3}{-4}$$

$$=-\frac{3}{4}$$

Step 3: Find the Equation of the Median

The equation of the median can be found using the point-slope form:

$$y-y_1=m(x-x_1)$$

where $(x_1,y_1)$ is a point on the line and $m$ is the slope.

In this case, we can use the point $(2,2)$ and the slope $-\frac{3}{4}$. Plugging in the values, we get:

$$y-2=-\frac{3}{4}(x-2)$$

Step 4: Simplify the Equation

To simplify the equation, we can multiply both sides by 4 to eliminate the fraction:

$$4(y-2)=-3(x-2)$$

Expanding and simplifying, we get:

$$4y-8=-3x+6$$

Adding 8 to both sides and adding 3x to both sides, we get:

$$4y+3x-14=0$$

Conclusion

In this article, we found the equation of the median of a triangle formed by three given points. We first found the midpoint of the opposite side, then found the slope of the median, and finally found the equation of the median using the point-slope form. The final equation of the median is:

$$3x+4y-14=0$$

This equation represents the line of symmetry of the triangle and divides the triangle into two equal areas.

References

• [1] "Geometry" by Michael Artin
• [2] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Keywords

• Median of a triangle
• Equation of a line
• Point-slope form
• Slope formula
• Midpoint formula
  Frequently Asked Questions (FAQs) about Finding the Equation of the Median of a Triangle =====================================================================================

Q: What is the median of a triangle?

A: The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. The median divides the triangle into two equal areas and is also the line of symmetry of the triangle.

Q: Why is the median of a triangle important?

A: The median of a triangle is important because it divides the triangle into two equal areas and is also the line of symmetry of the triangle. This makes it a useful tool for solving problems involving triangles.

Q: How do I find the equation of the median of a triangle?

A: To find the equation of the median of a triangle, you need to follow these steps:

1. Find the midpoint of the opposite side.
2. Find the slope of the median.
3. Use the point-slope form to find the equation of the median.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is:

$$y-y_1=m(x-x_1)$$

where $(x_1,y_1)$ is a point on the line and $m$ is the slope.

Q: How do I find the midpoint of a line segment?

A: To find the midpoint of a line segment, you can use the midpoint formula:

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

where $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of the two points.

Q: How do I find the slope of a line?

A: To find the slope of a line, you can use the slope formula:

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

where $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of the two points.

Q: What is the significance of the median of a triangle in real-life applications?

A: The median of a triangle has many real-life applications, such as:

• Architecture: The median of a triangle is used to design buildings and bridges.
• Engineering: The median of a triangle is used to design machines and mechanisms.
• Computer Science: The median of a triangle is used in algorithms and data structures.

Q: Can the median of a triangle be used to solve problems involving triangles?

A: Yes, the median of a triangle can be used to solve problems involving triangles. The median divides the triangle into two equal areas and is also the line of symmetry of the triangle, making it a useful tool for solving problems.

Q: What are some common mistakes to avoid when finding the equation of the median of a triangle?

A: Some common mistakes to avoid when finding the equation of the median of a triangle include:

• Not finding the midpoint of the opposite side correctly.
• Not finding the slope of the median correctly.
• Not using the point-slope form correctly.

Q: How can I practice finding the equation of the median of a triangle?

A: You can practice finding the equation of the median of a triangle by:

• Solving problems involving triangles.
• Using online resources and tools.
• Practicing with different types of triangles.

Conclusion

In this article, we have answered some frequently asked questions about finding the equation of the median of a triangle. We have covered topics such as the definition of the median of a triangle, the importance of the median, and how to find the equation of the median. We have also provided some common mistakes to avoid and some tips for practicing finding the equation of the median of a triangle.