Question

The triangle formed by the points (1, 5), (4, 2) and (–2, 2) is always a

• A. Right triangle
• B. Isosceles triangle
• C. Right-isosceles triangle
• D. Equilateral triangle

Answer 1

The correct option is C Right-isosceles triangle

Let the triangle formed by the points (1, 5), (4, 2) and (–2, 2) be ABC where A be (1, 5), B be (4, 2) and C be (–2, 2).

$AB=\sqrt{(4-1{)}^2+(2-5{)}^2}=3\sqrt{2}units$

$BC=\sqrt{(-2-4{)}^2+(2-2{)}^2}=6units$

$AC=\sqrt{(-2-1{)}^2+(2-5{)}^2}=3\sqrt{2}units$

Now,
$\begin{array}{cc}A{B}^2+A{C}^2& =3\sqrt{2}{)}^2+(3\sqrt{2}{)}^2\\ & =18+18\\ & =36\\ & =6^2\\ & =B{C}^2\end{array}$

Therefore, ΔABC is a right-angled at A.

Also, $AB=AC=3\sqrt{2}\text{units.}$

Thus, ΔABC is a right-isosceles triangle.

Hence, the correct answer is option (c).