Finding Triangle Vertex P Coordinates Using Centroid Formula
To find the coordinates of vertex P, we use the formula for the centroid of a triangle.
Centroid Formula
The coordinates of the centroid G(Gx, Gy) of a triangle with vertices P(x, y), Q(xQ, yQ), and R(xR, yR) are given by:
$G_x = \frac{x + x_Q + x_R}{3}$
$G_y = \frac{y + y_Q + y_R}{3}$
Applying the Formula
We are given:
- Vertex Q: (5, -1)
- Vertex R: (2, 6)
- Centroid G: (4, 2)
- Vertex P: (x, y)
Calculate X-coordinate of P
Using the formula for the x-coordinate of the centroid:
$4 = \frac{x + 5 + 2}{3}$
$4 = \frac{x + 7}{3}$
Multiply both sides by 3:
$12 = x + 7$
Solve for x:
$x = 12 - 7$
$x = 5$
Calculate Y-coordinate of P
Using the formula for the y-coordinate of the centroid:
$2 = \frac{y + (-1) + 6}{3}$
$2 = \frac{y + 5}{3}$
Multiply both sides by 3:
$6 = y + 5$
Solve for y:
$y = 6 - 5$
$y = 1$
Result
The coordinates of vertex P are (5, 1).
Similar Questions
- What is the equation of line through (1,2) and (3,8)
- A right triangle has vertices at P(0, 0), Q(8, 0), and R(0, 6). What are the coordinates of its Circumcenter?
- Find the x-intercept of $5x + 2y = 10$
- Point P is at an equal distance from the three vertices of a triangle with coordinates A(0, 0), B(4, 0), and C(0, 3). What are the coordinates of point P?
- Find slope of the line passing through points (-2,5) and (4,-7).
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Important Questions from Coordinate Geometry
What is the reflection of the point (-1, 5) in the line x = 1?
What are the co-ordinates of the centroid of a triangle, whose vertices are A(1, -5), B(-4, 0) and C(3, -4)?
Slope of the line AB is 4/3. Co-ordinates of points A and B are (x, -5) and (2, -3) respectively. What is the value of x?
Find the co-ordinates of the centroid of a triangle whose vertices are A(1, 4), B(7, 8) and C(10, 12).
- If x² + y² - 12x + 18y + 117 = 0, then the value of x² + y² is:
