Question

Triangles in which all the three sides are of equal length are called ............triangle.

Answer 1

Hint: We recall the definitions of side and vertex of a polygon. We recall the definition of length of a side triangle and then recall the definitions of scalene, isosceles, equilateral triangles based on relationship among the three sides of a triangle. We draw the figure of triangle ABC for scalene, isosceles, equilateral conditions on sides. We find that equilateral has three sides with equal length which means $AB=BC=AC$.

Complete step-by-step answer:
The closed curve which is formed only by joining a finite number of line segments is called a polygon and the point of intersection of line segments is called vertex. Any line segment which joins to form the polygon is called edge or side. We show two polygons. \[\]

We know that length of a line segment is distance between two endpoints. So the length of a side of a polygon is the distance between vertices joining the sides.\[\]

A triangle is a polygon with three sides and three vertices otherwise known 3-gon. If we denote the vertices of any triangle as A, B, C then its sides are denote as $\overline{AB},\overline{BC},\overline{AC}$ and length of the sides are denoted as $AB,BC,AC$. If the lengths of the triangle are not equal to each other which means $AB\ne BC\ne AC$ then it is called a scalene triangle. \[\]
 

If lengths of any two sides are equal we call the triangle isosceles triangle which means one of $AB=BC,BC=AC,AB=AC$ in triangle ABC is true. We draw the figure of a isosceles triangle with $AB=AC$.\[\]
 

If lengths of all sides of triangle are equal we call the triangle equilateral. The word ‘equilateral’ is formed from two Latin words aeqqus which means equal and lateralis which means belonging to the side. If ABC is an equilateral triangle then $AB=BC=AC$ whose figure we draw below\[\]
 

So the blank needs to be filled by the word ‘equilateral’. \[\]

Note: We note that the triangles with sides having equal length will have opposite angles with equal measure. So all the angles in an equilateral triangle will be equal and measure ${{60}^{\circ }}$. A polygon with equal sides is called a regular polygon and hence an equilateral triangle is called a regular triangle. If length of the side of an equilateral triangle is $a$ then its perimeter is $3a$ and the area is $\dfrac{\sqrt{3}}{4}{{a}^{2}}$.