there is a triangle with 3 points, A(4,5), B(1,2), C(6,2)

Question

To prove that all three medians of a triangle meet at the same point, the medians of each side must be found. What are the ordered pairs of the three median bisectors?

c=(1.5,1.5), b= (1,1.5), and a=(2.5,0)
c=(1,2), b=(4,5), and a=(6,2)
c=(3,4), b=(4.5, 4), and a=(4,2)
c=(2.5,3.5), b=(5,3.5), and a=(3.5,2)

Answer 1

To find the medians of triangle ABC with vertices at \( A(4,5) \), \( B(1,2) \), and \( C(6,2) \), we first need to determine the midpoints of each side of the triangle:

1. Midpoint of BC: \[ M_{BC} = \left( \frac{B_x + C_x}{2}, \frac{B_y + C_y}{2} \right) = \left( \frac{1 + 6}{2}, \frac{2 + 2}{2} \right) = \left( \frac{7}{2}, 2 \right) = (3.5, 2) \]

2. Midpoint of AC: \[ M_{AC} = \left( \frac{A_x + C_x}{2}, \frac{A_y + C_y}{2} \right) = \left( \frac{4 + 6}{2}, \frac{5 + 2}{2} \right) = \left( 5, \frac{7}{2} \right) = \left( 5, 3.5 \right) \]

3. Midpoint of AB: \[ M_{AB} = \left( \frac{A_x + B_x}{2}, \frac{A_y + B_y}{2} \right) = \left( \frac{4 + 1}{2}, \frac{5 + 2}{2} \right) = \left( \frac{5}{2}, \frac{7}{2} \right) = \left( 2.5, 3.5 \right) \]

Now, the medians of triangle ABC extend from each vertex to the corresponding midpoints we calculated.

• The median from vertex \( A \) to midpoint \( M_{BC} \) is the line segment from \( A(4,5) \) to \( M_{BC}(3.5, 2) \).
• The median from vertex \( B \) to midpoint \( M_{AC} \) is the line segment from \( B(1,2) \) to \( M_{AC}(5, 3.5) \).
• The median from vertex \( C \) to midpoint \( M_{AB} \) is the line segment from \( C(6,2) \) to \( M_{AB}(2.5, 3.5) \).

Ordered pairs of the midpoints of each side:

• The midpoint of \( BC \) is \( (3.5, 2) \).
• The midpoint of \( AC \) is \( (5, 3.5) \).
• The midpoint of \( AB \) is \( (2.5, 3.5) \).

From the options given, we can conclude that the correct medians connecting the vertices to the midpoints you are referring to do not directly match the values provided as the midpoints of the sides previously calculated were incorrect in understanding how the midpoints translate directly into the correct states.

The really median bisectors should connect to the midpoints calculated above. Thus the medians from \( A \), \( B \), and \( C \) connect to their respective midpoints \( M_{BC} \), \( M_{AC} \), and \( M_{AB} \).

Answer 2

it's a multiple choice question

Answer 3

I apologize for the confusion. Based on your request and the context of a multiple-choice question, let's accurately identify the midpoints of the sides and select the appropriate answer from the options provided.

We have vertices:

• \( A(4,5) \)
• \( B(1,2) \)
• \( C(6,2) \)

Midpoints:

1. Midpoint of BC: \[ M_{BC} = \left( \frac{1 + 6}{2}, \frac{2 + 2}{2} \right) = \left( \frac{7}{2}, 2 \right) = (3.5, 2) \]

2. Midpoint of AC: \[ M_{AC} = \left( \frac{4 + 6}{2}, \frac{5 + 2}{2} \right) = \left( 5, \frac{7}{2} \right) = \left( 5, 3.5 \right) \]

3. Midpoint of AB: \[ M_{AB} = \left( \frac{4 + 1}{2}, \frac{5 + 2}{2} \right) = \left( \frac{5}{2}, \frac{7}{2} \right) = \left( 2.5, 3.5 \right) \]

Evaluation of Options:

Based on the midpoints calculated:

• Midpoint of BC: \( (3.5, 2) \)
• Midpoint of AC: \( (5, 3.5) \)
• Midpoint of AB: \( (2.5, 3.5) \)

Now let's match these results to your provided options:

1. \( c=(1.5,1.5), b= (1,1.5), and a=(2.5,0) \)
2. \( c=(1,2), b=(4,5), and a=(6,2) \)
3. \( c=(3,4), b=(4.5, 4), and a=(4,2) \)
4. \( c=(2.5,3.5), b=(5,3.5), and a=(3.5,2) \)

Correct Option:

The ordered pairs of the three median bisectors, which match our calculations, are:

• \( c=(3.5, 2) \) (Midpoint of BC)
• \( b=(5, 3.5) \) (Midpoint of AC)
• \( a=(2.5, 3.5) \) (Midpoint of AB)

Thus the closest match from your options is: Option 4: \( c=(2.5,3.5), b=(5,3.5), and a=(3.5,2) \)

Please double-check the coordinates, as there may be a slight difference with the order of the points given for each median; however, it seems this may not fully represent the median bisectors stated.