If the coordinates of the mid points of sides of a triangle are (3,4) (4,6) and (5,7) find the vertices of the triangle? Please explain step by step.?

Question

Answer

Answer

Let the vertices of the triangle be A(x₁ , y₁ ), B(x₂ , y₂ ) and C(x₃ ,y₃ ).

Let the mid-points of the side BC be D (3, 4)

=> (x₂ + x₃/2, y₃ + y₂/2) = (3, 4)

=> x₂ + x₃ = 6,y₃ + y₂ = 8....(1)

Let the mid-points of the side AC be E (4, 6)

=>(x₁ + x₃/2, y₁ + y₃/2) = (4, 6)

=>x₁ + x₃ = 8, y₁ + y₃ = 12...(2)

Let the mid-points of the side AB be F (5, 7).

=> (x₁ + x₂/2, y₁ + y₂/2) = (5, 7)

=>x₁ + x₂ = 10, y₁ + y₂ = 14...(3)

We know that centroid of the triangle formed by the midpoints of the sides of the triangle will be same as the centroid of that triangle.

Hence, we get

G(x₁ + x₂ + x₃/3, y₁ + y₂ + y₃/3) = (4, 17/3)

=>x₁ + x₂ + x₃ = 12

y₁ + y₂ + y₃ = 17(*)

Now,

Using (*) and (1), we find that A(x₁, y₁) = (6, 9)

Using (*) and (2), we find that B(x₂, y₂) = (4, 5)

Using (*) and (3), we find that C(x₃, y₃) = (2, 3).

Answer

Given:
The coordinates of the midpoints of the sides of a triangle are (3,4), (4,6), and (5,7).

Approach:
To find the vertices of the triangle, we can use the concept that the midpoint of a side of a triangle divides the side in the ratio 1:2. We can find the lengths of the sides of the triangle using the distance formula and then use these lengths to find the coordinates of the vertices.

Step-by-step solution:

1. Find the lengths of the sides:
Let's consider the midpoints of the sides as A (3,4), B (4,6), and C (5,7). We need to find the lengths of the sides AB, BC, and AC.

The distance formula is given by:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Using the distance formula, we can calculate the lengths of the sides as follows:

- Length of side AB:
Distance AB = √((4 - 3)^2 + (6 - 4)^2)
= √(1^2 + 2^2)
= √(1 + 4)
= √5

- Length of side BC:
Distance BC = √((5 - 4)^2 + (7 - 6)^2)
= √(1^2 + 1^2)
= √(1 + 1)
= √2

- Length of side AC:
Distance AC = √((5 - 3)^2 + (7 - 4)^2)
= √(2^2 + 3^2)
= √(4 + 9)
= √13

2. Find the coordinates of the vertices:
Let the coordinates of the vertices be A(x1, y1), B(x2, y2), and C(x3, y3).

- Coordinates of A:
Since A is the midpoint of BC, the coordinates of A can be found using the midpoint formula:
x1 = (x2 + x3)/2
y1 = (y2 + y3)/2

Plugging in the given values, we get:
3 = (4 + 5)/2
4 = (6 + 7)/2

Solving these equations, we find that:
x1 = 9/2
y1 = 13/2

- Coordinates of B:
Using the same logic, the coordinates of B can be found:
x2 = (x1 + x3)/2
y2 = (y1 + y3)/2

Plugging in the given values, we get:
4 = (9/2 + 5)/2
6 = (13/2 + 7)/2

Solving these equations, we find that:
x2 = 7/2
y2 = 15/2

- Coordinates of C:
Using the same logic, the coordinates of C can be found:
x3 = (x1 + x2)/2
y3 = (y1 + y2)/2

Plugging in the given values, we get:
5 = (9/2 + 7/2)/2
7

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