Question

4i) which of the following transformations carry this regular polygon onto itself? image of a regular nonagon 4i) rotation of 60° counterclockwise rotation of 90° counterclockwise rotation of 36° counterclockwise rotation of 40° counterclockwise

Explanation:

Step1: Determine the number of sides

The polygon is a regular nonagon? Wait, no, looking at the shape, it's a regular nonagon? Wait, no, the number of sides: let's count the sides. Wait, the regular polygon here, let's see the rotation angle. For a regular polygon with \( n \) sides, the smallest rotation angle that maps it onto itself is \( \frac{360^\circ}{n} \). Let's check the options. The options are 60, 90, 36, 40. Let's find \( n \) such that \( \frac{360^\circ}{n} \) divides one of these angles. Let's check each:

• For 40°: \( \frac{360}{n}=40 \) → \( n = 9 \). Wait, 360/9=40? No, 360/9=40? Wait 940=360, yes. So if the polygon has 9 sides? Wait no, wait 360/9=40? Wait 940=360, yes. Wait, but let's check the shape. Wait, the polygon in the image: let's count the sides. Wait, the red polygon: let's see, the number of sides. Wait, maybe it's a nonagon? No, wait 360 divided by the rotation angle should be an integer (number of sides). Let's check each option:

• 60°: 360/60=6 → n=6 (hexagon). But the shape doesn't look like a hexagon.
• 90°: 360/90=4 → n=4 (square). No.
• 36°: 360/36=10 → n=10 (decagon). No.
• 40°: 360/40=9 → n=9 (nonagon). Wait, but the shape in the image: let's count the sides. Wait, maybe the polygon has 9 sides? Wait, no, maybe I miscounted. Wait, the image shows a regular polygon, let's check the rotation angle. The correct rotation that maps it onto itself must be a multiple of \( \frac{360^\circ}{n} \), where \( n \) is the number of sides. Let's check the options:

Wait, the correct answer is rotation of 40° counterclockwise? Wait, no, wait 360 divided by the number of sides. Let's see, if the polygon has 9 sides, then the rotation angle is 360/9=40°, so a rotation of 40° (or multiples) will map it onto itself. So the correct option is rotation of 40° counterclockwise.

Wait, let's recheck:

For a regular polygon with \( n \) sides, the rotational symmetry angle is \( \frac{360^\circ}{n} \). So if the rotation angle is a multiple of \( \frac{360^\circ}{n} \), it maps to itself.

Check each option:

• 60°: 360/60=6 → n=6 (hexagon). But the shape doesn't look like a hexagon (hexagon has 6 sides, this has more).
• 90°: 360/90=4 → square (4 sides). No.
• 36°: 360/36=10 → decagon (10 sides). No.
• 40°: 360/40=9 → nonagon (9 sides). So if the polygon has 9 sides, then 40° rotation (and multiples like 80°, 120°, etc.) will map it onto itself. So the correct option is rotation of 40° counterclockwise.

Step2: Verify the rotation angle

Since the regular polygon with 9 sides has a rotational symmetry of 40° (360/9=40), a rotation of 40° counterclockwise will carry it onto itself. The other angles (60, 90, 36) correspond to n=6,4,10, which don't match the polygon's side count. So the correct transformation is rotation of 40° counterclockwise.

Answer:

rotation of \( 40^\circ \) counterclockwise