Use the image to answer the question.

Question

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

(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
Skip to navigation

Answer 1

To find the coordinates of the third vertex of the triangle given the two vertices and the midpoint of one median, we will use the formula for the midpoint.

Let's denote the vertices of the triangle as follows:

• \( A(2, 5) \)
• \( B(5, 9) \)
• \( C(x, y) \) (the unknown vertex)

The median from vertex \( A \) (which has coordinates \( (2, 5) \)) intersects the midpoint of the line segment \( BC \) at the point \( (6.5, 6.5) \).

The coordinates of the midpoint \( M \) of the segment \( BC \) can be calculated using the formula:

\[ M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \]

For points \( B(5, 9) \) and \( C(x, y) \): \[ M = \left(\frac{5 + x}{2}, \frac{9 + y}{2}\right) \]

Since we know that the median intersects at point \( (6.5, 6.5) \), we can set these equal to find \( x \) and \( y \):

1. \(\frac{5 + x}{2} = 6.5\)
2. \(\frac{9 + y}{2} = 6.5\)

Now we solve these equations.

For the first equation: \[ 5 + x = 2 \times 6.5 = 13 \quad \Rightarrow \quad x = 13 - 5 = 8 \]

For the second equation: \[ 9 + y = 2 \times 6.5 = 13 \quad \Rightarrow \quad y = 13 - 9 = 4 \]

Thus, the coordinates of the third vertex \( C \) are \( (8, 4) \).

Therefore, the correct response is: (8, 4)