If the centroid of a triangle formed by (7, x), (y, -6) and (9, 10) is (6, 3), then the values of x and y are respectively
5, 2
Understanding the Centroid of a Triangle
The centroid of a triangle is a special point where the three medians of the triangle intersect. A median is a line segment that connects a vertex to the midpoint of the opposite side. The centroid represents the geometric center of the triangle.
If the vertices of a triangle are given as (x<sub>1</sub>, y<sub>1</sub>), (x<sub>2</sub>, y<sub>2</sub>), and (x<sub>3</sub>, y<sub>3</sub>), the coordinates of the centroid (G<sub>x</sub>, G<sub>y</sub>) can be found using the following formula:
\(G_x = \frac{x_1 + x_2 + x_3}{3}\)
\(G_y = \frac{y_1 + y_2 + y_3}{3}\)
Applying the Centroid Formula to Find Unknown Coordinates
In this problem, we are given the coordinates of the three vertices of a triangle and the coordinates of its centroid. Two of the coordinates of the vertices are unknown (x and y). We need to use the centroid formula to find the values of these unknowns.
The given information is:
- Vertex 1: (7, x)
- Vertex 2: (y, -6)
- Vertex 3: (9, 10)
- Centroid: (6, 3)
Let's plug these values into the centroid formula.
Finding the value of y using the x-coordinate of the Centroid
We use the formula for the x-coordinate of the centroid:
\(G_x = \frac{x_1 + x_2 + x_3}{3}\)
Substitute the given values:
\(6 = \frac{7 + y + 9}{3}\)
Now, we solve for y:
Multiply both sides by 3:
\(6 \times 3 = 7 + y + 9\)
\(18 = 16 + y\)
Subtract 16 from both sides:
\(18 - 16 = y\)
\(2 = y\)
So, the value of y is 2.
Finding the value of x using the y-coordinate of the Centroid
Next, we use the formula for the y-coordinate of the centroid:
\(G_y = \frac{y_1 + y_2 + y_3}{3}\)
Substitute the given values:
\(3 = \frac{x + (-6) + 10}{3}\)
Now, we solve for x:
Multiply both sides by 3:
\(3 \times 3 = x - 6 + 10\)
\(9 = x + 4\)
Subtract 4 from both sides:
\(9 - 4 = x\)
\(5 = x\)
So, the value of x is 5.
Result: Values of x and y
Based on our calculations using the centroid formula, we found that:
- x = 5
- y = 2
The question asks for the values of x and y respectively, which are 5 and 2.
| Coordinate | Calculation | Value |
|---|---|---|
| y (from G<sub>x</sub>) | \(\frac{7 + y + 9}{3} = 6 \implies 16 + y = 18 \implies y = 2\) | 2 |
| x (from G<sub>y</sub>) | \(\frac{x + (-6) + 10}{3} = 3 \implies x + 4 = 9 \implies x = 5\) | 5 |
Revision Table: Centroid Formula Review
| Concept | Description | Formula |
|---|---|---|
| Centroid | Point of intersection of medians; the geometric center of a triangle. | N/A |
| Median | Line segment from a vertex to the midpoint of the opposite side. | N/A |
| Centroid Coordinates | Average of the x-coordinates and average of the y-coordinates of the vertices. | \(G(x, y) = \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \right)\) |
Additional Information: Properties of the Centroid
The centroid is an important point in a triangle with several interesting properties:
- The centroid always lies inside the triangle.
- The centroid divides each median in a 2:1 ratio, with the longer segment being between the vertex and the centroid. For example, if A is a vertex and M is the midpoint of the opposite side, and G is the centroid, then AG : GM = 2 : 1.
- The centroid is also the center of mass of the triangle if the triangle is considered to have uniform density.
- The centroid is one of the four classical triangle centers, along with the incenter, circumcenter, and orthocenter.
Understanding the centroid formula and its properties is crucial for solving problems in coordinate geometry involving triangles.
Important Questions from Centroid
What is the area of the triangle formed by these lines?
The centroid of the triangle is at which one of the following points?
If \(\vec a, \vec b, \vec c\) , are the position vectors of the vertices A, B, C respectively of a triangle ABC and G is the centroid of the triangle, then what is \(\overrightarrow{AG}\) equal to ? .
The centroid of the triangle with vertices A(2, -3, 3), B(5, -3, -4) and C(2, -3, -2) is the point
If a vertex of a triangle is (1, 1) and the midpoints of two sides of the triangle through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is
