Question

Identify the number of convex polygons from the given figures.

Answer 1

Hint:Consider each set of three points along the polygon. If every angle is \[{180^ \circ }\] or less you have a convex polygon. When you figure out each angle, also keep a running total of \[{180^ \circ }\]angle. For a concave polygon, this will total \[{360^ \circ }\]
If a line segment joining any two points of the interior of a polygon lies inside the polygon then it is called convex polygon.
If each of the interior angles of a polygon is less than\[{180^ \circ }\], then it is called convex polygon.

Complete step-by-step answer:
It is given that we have to identify which of the given figures are convex polygons.
We also know that, if a line segment joining any two points of the interior of a polygon lies inside the polygon then it is called convex polygon.
Now let us check this condition for every figure
Initially let us check this condition for figure (2).
Let us consider figure 2, in that let us fix any two points on the shape clearly that points can be joined by a line segment in the interior of the shape.
Hence by the definition of convex polygon we can say that the polygon in figure (2) is a convex polygon.
Similarly on considering figure (6) any two points in the polygon have interior connection.
Every point in both the figures can be joined interior to the polygon given no two points require an exterior connection.
Whereas in all the other figures there are at least two points such that the line segment joining both cannot be joined inside the polygon. Such points are given in the following figures

These points do not have any interior connections therefore they are not convex polygons.
This condition is satisfied only in figure (2) and (6)
Hence we can conclude that the figure (2) and (6) are convex polygons.

Note: Diagonals lie interior for convex polygons and few diagonals lie outside for concave polygons.