First slide

Cartesian plane

Question

The points $A (2 a, 4 a), B (2 a, 6 a)$ and $C (2 a + \sqrt{3} a, 5 a)$ $(w h e n a > 0)$ are vertices of

Moderate

A

an obtuse angled triangle
B

an equilateral triangle
C

an isosceles obtuse angled triangle
D

a right angle triangle

Solution

$A B = \sqrt{{(2 a - 2 a)}^{2} + {(4 a - 6 a)}^{2}} = 2 a$
$B C = \sqrt{{(2 a + \sqrt{3} a - 2 a)}^{2} + {(5 a - 6 a)}^{2}} = \sqrt{{(\sqrt{3} a)}^{2} + a^{2}} = 2 a$ and

$C A = \sqrt{{(2 a - 2 a - \sqrt{3} a)}^{2} + {(5 a - 4 a)}^{2}} = \sqrt{{(\sqrt{3} a)}^{2} + a^{2}} = 2 a$

Since $A B = B C = C A,$ the triangle is an equilateral.

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