Math Problem Statement

If A(-1, 2), B(0, 0), and C(2, 1) are the vertices of ΔABC, then the length of the median through the vertex A is?

Solution

The question provided is:

If A(1,2)A(-1, 2), B(0,0)B(0, 0), and C(2,1)C(2, 1) are the vertices of ABC\triangle ABC, then the length of the median through the vertex AA is?


Solution:

To solve for the length of the median from vertex AA, we follow these steps:

  1. Find the midpoint of side BCBC:

The coordinates of BB are (0,0)(0, 0), and CC are (2,1)(2, 1).

The formula for the midpoint MM of a line segment with endpoints (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

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

Substitute the coordinates of B(0,0)B(0, 0) and C(2,1)C(2, 1):

M=(0+22,0+12)=(1,12)M = \left( \frac{0 + 2}{2}, \frac{0 + 1}{2} \right) = \left( 1, \frac{1}{2} \right)

Thus, the midpoint MM of BCBC is (1,12)\left( 1, \frac{1}{2} \right).


  1. Calculate the length of the median AMAM:

The length of a line segment between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:

d=(x2x1)2+(y2y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Here, A(1,2)A(-1, 2) and M(1,12)M\left( 1, \frac{1}{2} \right). Substituting these into the formula:

AM=(1(1))2+(122)2AM = \sqrt{\left( 1 - (-1) \right)^2 + \left( \frac{1}{2} - 2 \right)^2}

Simplify step by step:

  • 1(1)=1+1=21 - (-1) = 1 + 1 = 2
  • 122=1242=32\frac{1}{2} - 2 = \frac{1}{2} - \frac{4}{2} = -\frac{3}{2}

Now square these values:

  • 22=42^2 = 4
  • (32)2=94\left( -\frac{3}{2} \right)^2 = \frac{9}{4}

Add these together:

AM=4+94AM = \sqrt{4 + \frac{9}{4}}

Convert 44 to have a denominator of 44:

4=1644 = \frac{16}{4}

Thus:

AM=164+94=254AM = \sqrt{\frac{16}{4} + \frac{9}{4}} = \sqrt{\frac{25}{4}}

Simplify:

AM=254=52AM = \frac{\sqrt{25}}{\sqrt{4}} = \frac{5}{2}


Final Answer:

The length of the median through vertex AA is:

52units\boxed{\frac{5}{2} \, \text{units}}


Do you want me to explain any specific step further? Here are related questions you can explore:

  1. What is the formula for a median of a triangle?
  2. How do you find the centroid of a triangle using coordinates?
  3. How would the problem change if another vertex DD was introduced?
  4. What is the significance of a median in geometry?
  5. How would you calculate the perimeter of ABC\triangle ABC?

Tip: The midpoint formula is essential in coordinate geometry for dividing a line segment into equal halves!

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Triangle Properties
Median of a Triangle

Formulas

Midpoint Formula: M = ((x1 + x2)/2, (y1 + y2)/2)
Distance Formula: d = √((x2 - x1)^2 + (y2 - y1)^2)

Theorems

Properties of Median in a Triangle

Suitable Grade Level

Grades 9-11