Question

Use the image to answer the 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.

a=(4,5),b=(1,2),c=(6,2)

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 a=(3.5,2)
c equals left parenthesis 2.5 comma 3.5 right parenthesis , b equals left parenthesis 5 comma 3.5 right parenthesis , and a equals left parenthesis 3.5 comma 2 right parenthesis

Answer 1

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

1. **Find the midpoint of side BC:**
- B = (1, 2) and C = (6, 2).
- Midpoint of BC = \(\left(\frac{1 + 6}{2}, \frac{2 + 2}{2}\right) = \left(\frac{7}{2}, 2\right) = (3.5, 2)\).

2. **Find the midpoint of side AC:**
- A = (4, 5) and C = (6, 2).
- Midpoint of AC = \(\left(\frac{4 + 6}{2}, \frac{5 + 2}{2}\right) = \left(5, \frac{7}{2}\right) = (5, 3.5)\).

3. **Find the midpoint of side AB:**
- A = (4, 5) and B = (1, 2).
- Midpoint of AB = \(\left(\frac{4 + 1}{2}, \frac{5 + 2}{2}\right) = \left(\frac{5}{2}, \frac{7}{2}\right) = (2.5, 3.5)\).

Now we can summarize the midpoints (which represent the median bisectors):

- Midpoint of BC = (3.5, 2)
- Midpoint of AC = (5, 3.5)
- Midpoint of AB = (2.5, 3.5)

Based on your options, the correct answer corresponds with

- c = (3.5, 2)
- b = (5, 3.5)
- a = (2.5, 3.5)

So the answer is:

**c = (3.5, 2), b = (5, 3.5), and a = (2.5, 3.5)**.

Answer 2

Triangle ABC
is drawn on a coordinate plane with vertices A(−3,−3)
, B(0,6)
, and C(4,−3)
and medians indicating the midpoint of each of the lines AB
, BC
, and CA
. Prove the medians meet at a single point by finding the centroid.(1 point)
Responses

(0,13)
left parenthesis 0 comma Start Fraction 1 over 3 End Fraction right parenthesis

(16,3)
left parenthesis Start Fraction 1 over 6 End Fraction comma 3 right parenthesis

(13,0)
left parenthesis Start Fraction 1 over 3 End Fraction comma 0 right parenthesis

(13,1)

Answer 3

To find the centroid of triangle ABC with vertices A(-3, -3), B(0, 6), and C(4, -3), we can use the formula for the centroid (G) of a triangle given its vertices (x₁, y₁), (x₂, y₂), (x₃, y₃):

\[
G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]

Substituting the coordinates of the vertices:

- A = (-3, -3)
- B = (0, 6)
- C = (4, -3)

We can calculate:

1. **Sum of the x-coordinates:**
\[
x_1 + x_2 + x_3 = -3 + 0 + 4 = 1
\]

2. **Sum of the y-coordinates:**
\[
y_1 + y_2 + y_3 = -3 + 6 - 3 = 0
\]

Now, substitute these sums into the centroid formula:

\[
G = \left( \frac{1}{3}, \frac{0}{3} \right) = \left( \frac{1}{3}, 0 \right)
\]

Now let's check the responses provided:

1. (0, 13)
2. \(\left(0, \frac{1}{3}\right)\)
3. \((13, 0)\)
4. (13, 1)

The correct answer is:

**\(\left( \frac{1}{3}, 0 \right)\)**.