Use the image to answer the question.

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:

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