Question

The points $(a,a),(–a,-a)$ and $(–\sqrt{3}a,\sqrt{3}a)$ are the vertices of a/an _____ triangle.

• A. scalene
• B. right angled
• C. isosceles right angled
• D. equilateral

Answer 1

The correct option is D

equilateral

Let's name the points as $A(a,a),B(-a,-a)$ and $C(–\sqrt{3}a,\sqrt{3}a)$.

By distance formula, distance between points $(x_1,y_1)$ and $(x_2,y_2)$

$=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

$BC=\sqrt{(-\sqrt{3}a+a{)}^2+(\sqrt{3}a+a{)}^2}=\sqrt{a^2+(\sqrt{3}a{)}^2-2\times a\times \sqrt{3}a+\sqrt{3}a{)}^2+a^2+2\times a\times \sqrt{3}a}BC=\sqrt{8a^2}=2\sqrt{2}a\text{units}\text{Similarly}AB=\sqrt{(a+a{)}^2+(a+a{)}^2}=\sqrt{8a^2}=2\sqrt{2}a\text{units}AC=\sqrt{(-\sqrt{3}a-a{)}^2+(\sqrt{3}a-a{)}^2}=\sqrt{8a^2}=2\sqrt{2}a\text{units}$

$\text{We get,}AB=BC=AC=2\sqrt{2}a\text{units}\therefore △\text{ABC is an equilateral triangle}.$