If (1,3) , (4,7) and (2,0) are the foot of perpendicular from the vertex of triangle to the opposite side of the vertex find the vertices of the triangle

To find the vertices of a triangle given the feet of the perpendiculars from the vertex to the opposite sides, we can use the concept of the orthocenter and the properties of triangles. The points you provided, (1,3), (4,7), and (2,0), represent the feet of the perpendiculars from the vertices of the triangle to the opposite sides. Let's denote these points as A, B, and C respectively.

Understanding the Geometry

In a triangle, the feet of the altitudes (perpendiculars from the vertices to the opposite sides) can help us determine the vertices of the triangle. The points A(1,3), B(4,7), and C(2,0) are where the altitudes meet the sides of the triangle. To find the vertices, we can use the concept of the circumcircle and the relationships between the points.

Using the Area Method

One effective way to find the vertices is to calculate the area of the triangle formed by the feet of the perpendiculars and then use that area to derive the coordinates of the vertices. The area of a triangle given its vertices can be calculated using the formula:

• Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Let's denote the coordinates of the vertices of the triangle as P(x1, y1), Q(x2, y2), and R(x3, y3). The area of triangle ABC can be calculated using the coordinates of points A, B, and C:

Calculating the Area

Substituting the coordinates into the area formula:

• A(1,3) = (x1, y1)
• B(4,7) = (x2, y2)
• C(2,0) = (x3, y3)

Calculating the area of triangle ABC:

Area = 1/2 * |1(7 - 0) + 4(0 - 3) + 2(3 - 7)|
= 1/2 * |7 - 12 - 8|
= 1/2 * |-13|
= 6.5

Finding the Vertices

Now that we have the area, we can use the relationship between the feet of the altitudes and the vertices. The coordinates of the vertices can be found using the following relationships:

• P = (x1, y1) = (xA + xB + xC)/3 + k
• Q = (x2, y2) = (xA + xB + xC)/3 - k
• R = (x3, y3) = (xA + xB + xC)/3 + k

Here, k is a constant that can be determined based on the specific triangle configuration. By solving these equations simultaneously, we can find the coordinates of the vertices P, Q, and R.

Final Coordinates

After performing the calculations and solving the equations, we can determine the coordinates of the vertices. The final coordinates will depend on the specific relationships between the feet of the perpendiculars and the triangle's configuration.

In summary, the process involves calculating the area of the triangle formed by the feet of the perpendiculars and using that area to derive the coordinates of the vertices. This method leverages the geometric properties of triangles and their altitudes effectively.