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A coordinate plane shows two segments. Both the axes range from 0 to 10 labeled in increments of 2.

To prove that the medians of a triangle all intersect at the same point, it’s first necessary to identify the vertices of the triangle. Rube draws a triangle on the coordinate plane. Two of the vertices of the triangle are located at (2,5)
and (5,9)
. The median that runs from the vertex (2,5)
ends at the point (6.5,6.5)
. What are the coordinates of the third vertex of the triangle?

(1 point)
Responses
which is the true correct answer and why
(8,4)
left parenthesis 8 comma 4 right parenthesis

(5.8,7.8)
open paren 5 point 8 comma 7 point 8 close paren

(6.5,6.5)
left parenthesis 6.5 comma 6.5 right parenthesis

(4,8)
left parenthesis 4 comma 8 right parenthesis
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1 answer

To find the third vertex of the triangle, we can use the properties of the median and the midpoint of the opposite side.

We have two vertices of the triangle: \( A(2, 5) \) and \( B(5, 9) \), and the median from vertex \( A \) ends at the point \( M(6.5, 6.5) \). The point \( M \) is the midpoint of the line segment connecting the third vertex \( C \) and the midpoint of \( AB \).

First, we calculate the midpoint \( D \) of the segment \( AB \): \[ D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 5}{2}, \frac{5 + 9}{2} \right) = \left( \frac{7}{2}, 7 \right) = (3.5, 7) \]

Next, since \( M \) is the midpoint of segment \( AC \), we can set up the midpoint formula for point \( M \): \[ M = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) \] Substituting \( A(2, 5) \) and \( M(6.5, 6.5) \): \[ (6.5, 6.5) = \left( \frac{2 + x_C}{2}, \frac{5 + y_C}{2} \right) \]

From this, we can create two equations:

  1. \( 6.5 = \frac{2 + x_C}{2} \)
  2. \( 6.5 = \frac{5 + y_C}{2} \)

Now, solving for \( x_C \) and \( y_C \):

  1. For \( x_C \): \[ 6.5 \cdot 2 = 2 + x_C \ 13 = 2 + x_C \ x_C = 13 - 2 \ x_C = 11 \]

  2. For \( y_C \): \[ 6.5 \cdot 2 = 5 + y_C \ 13 = 5 + y_C \ y_C = 13 - 5 \ y_C = 8 \]

Thus, the coordinates of the third vertex \( C \) are \( (11, 8) \).

However, none of the responses list the correct point \( (11, 8) \). Let's review the given options to find the best match:

  • \( (8, 4) \)
  • \( (5.8, 7.8) \)
  • \( (6.5, 6.5) \)
  • \( (4, 8) \)

Among the provided options, \( (4, 8) \) is the closest possible point when considering geometric positions and median properties, but it still does not match the calculated position.

Conclusively, based on the information and calculations provided, none of the listed points accurately represents the third vertex of the triangle. The third vertex should be at \( (11, 8) \).