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 \(90^{\circ}\) or concave if the internal angle is \(270^{\circ}\). If the number of convex corners in such a polygon is 25, the number of concave corners must be

• A. 20
• B. 0
• C. 21
• D. 22

Hint:

Write the sum of the interior angles in two ways, once from the number of corners and once from the 90 and 270 degree corners. Compare the two.

Answer 1

The Correct Option is C

Step 1: Understand the polygon:
Every side is parallel to the X axis or the Y axis, so at every corner the boundary turns by a right angle. The inside angle at a corner is either \(90^{\circ}\) (convex) or \(270^{\circ}\) (concave).
Let \(c\) be the number of convex corners and \(k\) the number of concave corners. We are told \(c = 25\) and we need \(k\).

Step 2: Use the angle sum of a polygon:
A polygon with \(n\) corners has interior angles adding up to \((n - 2) \times 180^{\circ}\). Here \(n = c + k\).
The interior angles also add up to \(90c + 270k\) degrees, because each convex corner gives 90 and each concave corner gives 270.

Step 3: Set the two sums equal:
\[ 90c + 270k = 180(c + k - 2) \]
\[ 90c + 270k = 180c + 180k - 360 \]
\[ 90k + 360 = 90c \]
Divide by 90:
\[ c - k = 4 \]

Step 4: Put in the given number:
With \(c = 25\):
\[ k = 25 - 4 = 21 \]
So the polygon has 21 concave corners.

Step 5: Check the options:
Option 1 (20) would give \(c - k = 5\), which breaks the rule \(c - k = 4\).
Option 2 (0) means no concave corners, then \(c = 4\) (a rectangle), not 25.
Option 4 (22) would give \(c - k = 3\), again wrong.
Only 21 gives \(c - k = 4\). A quick picture agrees: a plain rectangle has 4 convex and 0 concave corners, and every notch cut into the edge adds one concave and one convex corner, which keeps the difference at 4.

Final Answer:
The number of concave corners is 21, which is option 3. \[ \boxed{\text{Option 3: } 21} \]