Question

2. The midpoints of the sides of a triangle are (5, 1), (3,-5) ad (-5, -1). Find the co-ordinates of the vertices of the triangle.​

Answer 1

Given,

• The co-ordinate of vertices of the Triangle are (5, 1), (3, -5) and (-5, -1).

To Find,

• The co-ordinates of the vertices of the Triangle which is made on the Mid - points of given Triangle.

Solution,

Let the A (5, 1), B (3, -5) and C (-5, -1)

And, The Mid - points of sides are P, Q and R respectively.

Finding the Co-ordinate of P,

Mid - point Formula,

• $\boxed{x,y = \bigg[ \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg] }$

Here,

• $x_1 = 5, \: \: x_2 = 3, \: \: y_1 = 1, \: \: \text{and} \: \: y_2 = - 5$

Substituting the values,

$:\longmapsto\text{P}(x,y) = \bigg[ \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg] \\ \\ :\longmapsto\text{P}(x,y) = \frac{5 + 3}{2} , \frac{1 - 5}{2} \\ \\ :\longmapsto\text{P}(x,y) = \frac{8}{2}, \frac{ - 4}{2} \\ \\ \underline{:\longmapsto \text{P}(x,y) =(4, - 2)}$

Hence,

• The Co-ordinate of P is (4, -2).

.

Finding the Co-ordinate of Q,

Here,

• $x_1 = 3, \: \: x_2 = - 5, \: \: y_1 = - 5, \: \: \text{and} \: \: y_2 = - 1$

Substituting the values,

$:\longmapsto\text{Q}(x,y) = \bigg[ \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg] \\ \\ :\longmapsto\text{Q}(x,y) = \frac{3 - 5 }{2} , \frac{ - 5 - 1}{2} \\ \\ :\longmapsto\text{Q}(x,y) = \frac{ - 2}{2}, \frac{ - 6}{2} \\ \\ \underline{:\longmapsto \text{Q}(x,y) =( - 1, - 3)}$

Hence,

• The Co-ordinate of Q is (-1, -3).

.

Finding the Co-ordinate of Q,

Here,

• $x_1 = - 5, \: \: x_2 = 5, \: \: y_1 = - 1, \: \: \text{and} \: \: y_2 = 1$

Substituting the values,

$:\longmapsto\text{R}(x,y) = \bigg[ \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg] \\ \\ :\longmapsto\text{R}(x,y) = \frac{ - 5 + 5 }{2} , \frac{ - 1 + 1}{2} \\ \\ :\longmapsto\text{R}(x,y) = \frac{ 0}{2}, \frac{ 0}{2} \\ \\ \underline{:\longmapsto \text{R}(x,y) =( 0, 0)}$

Hence,

• The Co-ordinate of R is (0, 0).

Required Answer,

• The Co-ordinate of Vertices of the Triangle which is made on the Mid - point of given Triangle is (4, -2), (-1, -3) and (0, 0).