Question

If a vertex of a triangle is (1, 1) and the mid points of two sides through this vertex are (-1, 2) and (3, 2), then what type of triangle is this?

• A. Equilateral triangle
• B. Isoscales triangles
• C. Right angle triangle
• D. None of these

Answer 1

The correct option is B

Isoscales triangles

Let $(x_1,y_1)$ and $(x_2,y_2)$ be the other two vertices of the triangle. (-1, 2) be the mid point of vertices (1, 1) & $(x_1,y_1)$

$\frac{x_1+1}{2}=-1,\frac{y_1+1}{2}=2$

$x_1$ = -2 -1 = -3, $y_1$ = 4 - 1 = 3

$(x_1,y_1)$ = (-3, 3)

Similarly, (3, 2) be the midpoint of the vertices (1, 1) and $x_2,y_2$

$\frac{1+x_2}{2}=3,\frac{1+y_2}{2}=2$

$x_2$ = 5, $y_2$ = 3

$(x_2,y_2)$ = (5, 3)

Points of the vertices are A(1, 1), B(-3, 3), C(5, 3)

side AB = $\sqrt{(-3-1{)}^2+(3-1{)}^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}=4.472$

BC = $\sqrt{(5+3{)}^2+(3-3{)}^2}=\sqrt{(8{)}^2}=8$

CA = $\sqrt{(5-1{)}^2+(3-1{)}^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}=4.472$

Since, sum of the two sides is more than third side these 3 points are the vertices of a triangle

Two sides are equal AB and CA.

So, these points are the vertices of an isosceles triangle.