Question

Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a
polygon is said to be convex if the internal angle is 90° or concave if the internal angle is
270°. If the number of convex corners in such a polygon is 25, the number of concave
corners must be
A..21
B.22
C.23
D.24​

Answer 1

Given :-

• Each side of a given polygon is parallel to either the X or the Y axis.
• A corner of such a polygon is said to be convex if the internal angle is 90° or concave if the internal angle is 270°.
• The number of convex corners in such a polygon is 25.

To Find :-

The number of concave corners must be :-

A..21

B.22

C.23

D.24

Solution :-

Let us assume that, the total number of 90° angle is x and the total number of 270° angle is y.

Than,

→ sum of all 90° angles = 90 * x = 90x

→ sum of all 270° angles = 270 * y = 270y .

and,

→ Total sides polygon = (x + y).

Now, we know that,

• sum of all interior angles of a polygon with n sides is equal to = (n - 2) * 180° .

So,

→ sum of all angles of polygon with total sides (x + y) = [x + y - 2] * 180° .

Therefore,

→ 90x + 270y = (x + y - 2) * 180°

→ 90x + 270y = 180x + 180y - 360°

→ 180x - 90x + 180y - 270y = 360°

→ 90x - 90y = 360°

→ 90(x - y) = 360°

dividing both sides by 90°

→ x - y = 4

Putting given value of x = 25 now,

→ 25 - y = 4

→ y = 25 - 4

→ y = 21. (Option A) (Ans.)

Hence, the number of concave corners must be 21.