MCQ
If the vertices of a triangle have integral coordinates, then the triangle is
- AEquilateral
- ✓Never equilateral
- CIsosceles
- DNone of these
✓
Answer
Correct option: B.
Never equilateral
b
(b) Let and be the coordinates of the vertices of a triangle and let be all integers. Then the area of triangle
= Some rational numbers, because $x’s$ and $y's$ are integers.
Also, if the triangle is equilateral and a be the length of its side, then ${a^2} = {({x_1} - {x_2})^2} + {({y_1} - {y_2})^2} = A$ positive integer.
(b) Let and be the coordinates of the vertices of a triangle and let be all integers. Then the area of triangle
= Some rational numbers, because $x’s$ and $y's$ are integers.
Also, if the triangle is equilateral and a be the length of its side, then ${a^2} = {({x_1} - {x_2})^2} + {({y_1} - {y_2})^2} = A$ positive integer.
The area of the triangle $ = \frac{1}{2}bc\,\,\sin A$ $ = \frac{1}{2}.\,a\,.\,a\,\sin {60^o} = \frac{{\sqrt 3 }}{4}{a^2}$every angle is of ${60^o}\} $
Which is irrationals, because ${a^2}$ is positive integer.
But earlier we have calculated that the area of the triangle is rational number. Hence it is a contradiction. Therefore, if the vertices of a triangle are integers, the triangle cannot be equilateral.
Note : Students should remember this question as a fact.
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