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Triangle ABC

is drawn on a coordinate plane with vertices A(−2,−3)
, B(4, 0)
, and C(−2, 2)
and medians indicating the midpoint of each of the line segments AB¯¯¯¯¯¯¯¯
, BC¯¯¯¯¯¯¯¯
, and CA¯¯¯¯¯¯¯¯
. Prove that the medians meet at a single point by finding the centroid. (1 point)
Responses

(−1, 53)
open paren negative 1 comma 5 thirds close paren

(0, 0)
open paren 0 comma 0 close paren

(0,−13)
open paren 0 comma negative 1 third close paren

(0, 53)
open paren 0 comma 5 thirds close paren

5 answers

To find the centroid of triangle ABC, we can use the formula for the centroid (G) of a triangle with vertices at coordinates A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃). The centroid's coordinates are given by:

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

Using the vertices of triangle ABC:

  • A(−2,−3)
  • B(4, 0)
  • C(−2, 2)

We can substitute these values into the formula:

  1. Calculate \(x_G\): \[ x_G = \frac{-2 + 4 - 2}{3} = \frac{0}{3} = 0 \]

  2. Calculate \(y_G\): \[ y_G = \frac{-3 + 0 + 2}{3} = \frac{-1}{3} \]

Thus, the coordinates of the centroid \(G\) are: \[ G\left(0, -\frac{1}{3}\right) \]

Now, we can look at the options provided in the responses to see which matches:

  • (−1, 53) → not matching
  • (0, 0) → not matching
  • (0,−13) → not matching
  • (0, 53) → not matching

It appears there was a miscalculation in my last step, let's go over the math for \(y_G\) again.

We need to calculate:

\[ y_G = \frac{-3 + 0 + 2}{3} = \frac{-1}{3} \]

And the centroid is indeed \( G\left(0, -\frac{1}{3}\right) \).

None of the options provided correctly list the coordinates of the centroid we computed as \( G(0, -\frac{1}{3})\).

Therefore, while we have confirmed that the medians of the triangle meet at a single point (the centroid), it appears that the correct response isn't provided in the options listed.

remove the extra symbols and do not change the answer

To find the centroid of triangle ABC, we can use the formula for the centroid of a triangle with vertices at coordinates A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃). The centroid's coordinates are given by:

G(x_G, y_G) = ( (x₁ + x₂ + x₃) / 3 , (y₁ + y₂ + y₃) / 3 )

Using the vertices of triangle ABC:

  • A(−2,−3)
  • B(4, 0)
  • C(−2, 2)

We substitute these values into the formula:

  1. Calculate x_G: x_G = (−2 + 4 − 2) / 3 = 0 / 3 = 0

  2. Calculate y_G: y_G = (−3 + 0 + 2) / 3 = −1 / 3

Thus, the coordinates of the centroid G are: G(0, −1/3)

Therefore, the medians of the triangle meet at a single point, confirmed to be G(0, −1/3). None of the options provided correctly list these coordinates.

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

To find the ordered pairs of the medians of triangle ABC with vertices A(4,5), B(1,2), and C(6,2), we need to calculate the midpoints of the sides opposite each vertex and then use those midpoints to find the equations of the medians.

  1. Find the midpoints:

    • Midpoint between vertices B(1,2) and C(6,2): \[ M_{BC} = \left( \frac{1 + 6}{2}, \frac{2 + 2}{2} \right) = \left( \frac{7}{2}, 2 \right) = (3.5, 2) \]

    • Midpoint between vertices A(4,5) and C(6,2): \[ M_{AC} = \left( \frac{4 + 6}{2}, \frac{5 + 2}{2} \right) = \left( \frac{10}{2}, \frac{7}{2} \right) = (5, 3.5) \]

    • Midpoint between vertices A(4,5) and B(1,2): \[ M_{AB} = \left( \frac{4 + 1}{2}, \frac{5 + 2}{2} \right) = \left( \frac{5}{2}, \frac{7}{2} \right) = (2.5, 3.5) \]

  2. Determine the median endpoints:

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

Now, let’s summarize:

  • Median from A(4,5) to M_{BC}(3.5, 2)
  • Median from B(1,2) to M_{AC}(5, 3.5)
  • Median from C(6,2) to M_{AB}(2.5, 3.5)

Therefore, the ordered pairs of the three median bisectors are:

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

After checking which of the provided response options matches these coordinates:

The closest match appears to be:

  • \( c = (5, 3.5), b = (3.5, 2), a = (2.5, 3.5) \)

Choose the response: c = (5, 3.5), b = (3.5, 2), and a = (2.5, 3.5).

(Note: Ensure you select the correct numbers indicated in the options provided, as they might differ while listing.)