Question

Use the image to answer the 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

(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 of triangle ABC, we start with the coordinates of vertices A and B and use the formula for the centroid of a triangle.

The formula for the centroid (G) of a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by:

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

In this case, we have:
- Vertex A at (3, 8)
- Vertex B at (3, 2)
- Centroid P at (4, 5)

Let the coordinates of vertex C be (x₃, y₃).

Now we can set up the equations for the centroid:

1. For the x-coordinate of the centroid:
\[
\frac{3 + 3 + x_3}{3} = 4
\]

Multiplying both sides by 3 gives:
\[
3 + 3 + x_3 = 12
\]
\[
6 + x_3 = 12
\]
\[
x_3 = 12 - 6 = 6
\]

2. For the y-coordinate of the centroid:
\[
\frac{8 + 2 + y_3}{3} = 5
\]

Multiplying both sides by 3 gives:
\[
8 + 2 + y_3 = 15
\]
\[
10 + y_3 = 15
\]
\[
y_3 = 15 - 10 = 5
\]

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

Therefore, the answer is:
\[
\text{(6, 5)}
\]