Question
A coordinate plane's axes range from 0 to 8, both by 1-unit increments. A triangle made up of vertices upper A, upper B, and upper C is plotted. The coordinates of the vertices are given.
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?
(1 point)
Responses
c=(1,2), b=(4,5), and a=(6,2)
c equals left parenthesis 1 comma 2 right parenthesis , b equals left parenthesis 4 comma 5 right parenthesis , and a equals left parenthesis 6 comma 2 right parenthesis
c=(3,4), b=(4.5,4), and a=(4,2)
c equals left parenthesis 3 comma 4 right parenthesis , b equals left parenthesis 4.5 comma 4 right parenthesis , and a equals left parenthesis 4 comma 2 right parenthesis
c=(1.5,1.5), b=(1,1.5), and a=(2.5,0)
c equals left parenthesis 1.5 comma 1.5 right parenthesis , b equals left parenthesis 1 comma 1.5 right parenthesis , and a equals left parenthesis 2.5 comma 0 right parenthesis
c=(2.5,3.5), b=(5,3.5), and
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?
(1 point)
Responses
c=(1,2), b=(4,5), and a=(6,2)
c equals left parenthesis 1 comma 2 right parenthesis , b equals left parenthesis 4 comma 5 right parenthesis , and a equals left parenthesis 6 comma 2 right parenthesis
c=(3,4), b=(4.5,4), and a=(4,2)
c equals left parenthesis 3 comma 4 right parenthesis , b equals left parenthesis 4.5 comma 4 right parenthesis , and a equals left parenthesis 4 comma 2 right parenthesis
c=(1.5,1.5), b=(1,1.5), and a=(2.5,0)
c equals left parenthesis 1.5 comma 1.5 right parenthesis , b equals left parenthesis 1 comma 1.5 right parenthesis , and a equals left parenthesis 2.5 comma 0 right parenthesis
c=(2.5,3.5), b=(5,3.5), and
Answers
GPT-4o mini
To find the ordered pairs of the three median bisectors of triangle ABC, we first need the coordinates of the vertices A, B, and C. However, you did not provide the coordinates of these vertices in your question.
The medians of a triangle are the line segments that connect each vertex to the midpoint of the opposite side. For each side of the triangle, you calculate the midpoint and then find the equation of the median line.
1. **Find the midpoints of each side**:
- **Midpoint M_A** of side BC = \( \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right) \)
- **Midpoint M_B** of side AC = \( \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) \)
- **Midpoint M_C** of side AB = \( \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) \)
2. **Find the equations of the medians**:
- The median from vertex A to midpoint M_A (along the line connecting A to M_A).
- The median from vertex B to midpoint M_B.
- The median from vertex C to midpoint M_C.
3. **Equations of the medians would yield intersection point - centroid**:
- Solve the equations of the three medians to show they intersect at a single point.
If you provide the coordinates of vertices A, B, and C, I can help you determine the medians and their intersection point more accurately!
The medians of a triangle are the line segments that connect each vertex to the midpoint of the opposite side. For each side of the triangle, you calculate the midpoint and then find the equation of the median line.
1. **Find the midpoints of each side**:
- **Midpoint M_A** of side BC = \( \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right) \)
- **Midpoint M_B** of side AC = \( \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) \)
- **Midpoint M_C** of side AB = \( \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) \)
2. **Find the equations of the medians**:
- The median from vertex A to midpoint M_A (along the line connecting A to M_A).
- The median from vertex B to midpoint M_B.
- The median from vertex C to midpoint M_C.
3. **Equations of the medians would yield intersection point - centroid**:
- Solve the equations of the three medians to show they intersect at a single point.
If you provide the coordinates of vertices A, B, and C, I can help you determine the medians and their intersection point more accurately!