Use the image to answer the question.

Question

A coordinate plane shows three data points. Both the axes range from 0 to 10 labeled in increments of 2.

A’linda draws triangle ABC
on the coordinate plane. Vertex A
of the triangle is located at (3,8)
and vertex B
is located at (3,2)
. The centroid of the figure is indicated by the point P
, which is located at (4,5)
. What are the coordinates of C
, the third vertex of the triangle?

(1 point)
Responses which answer is correct and why

(6,8)
left parenthesis 6 comma 8 right parenthesis

(5,5)
left parenthesis 5 comma 5 right parenthesis

(3,5)
left parenthesis 3 comma 5 right parenthesis

(6,5)
left parenthesis 6 comma 5 right parenthesis
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Answer 1

To find the coordinates of vertex C in triangle ABC, we can use the properties of the centroid of a triangle. The centroid (point P) of a triangle can be found using the formula:

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

where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of the vertices A, B, and C, respectively.

We are given:

• Vertex A at \( (3, 8) \)
• Vertex B at \( (3, 2) \)
• Centroid P at \( (4, 5) \)

Let the coordinates of vertex C be \( (x, y) \). Using the centroid formula:

\[ P_x = \frac{x_1 + x_2 + x_3}{3} \] \[ 4 = \frac{3 + 3 + x}{3} \]

Multiplying both sides by 3:

\[ 12 = 3 + 3 + x \]

Simplifying:

\[ 12 = 6 + x \]

Solving for \( x \):

\[ x = 12 - 6 = 6 \]

Now for the y-coordinate:

\[ P_y = \frac{y_1 + y_2 + y_3}{3} \] \[ 5 = \frac{8 + 2 + y}{3} \]

Again, multiplying both sides by 3:

\[ 15 = 8 + 2 + y \]

Simplifying:

\[ 15 = 10 + y \]

Solving for \( y \):

\[ y = 15 - 10 = 5 \]

Thus, the coordinates of vertex C are \( (6, 5) \).

So, the correct answer is:

(6, 5).