Question

hwk #2 09/11/25 • draw 4 inscribed 7 - sided polygons in a circle radius 5 cm (heptagons) • draw 1 equilateral triangle in a 6 cm radius circle

Explanation:

Part 1: Drawing 4 Inscribed 7 - Sided Polygons (Heptagons) in a Circle of Radius 5 cm

Step 1: Prepare the Compass

Set the compass to a radius of \( r = 5\space\text{cm} \). Draw a circle with this radius. This circle will be the circum - circle for all the heptagons.

Step 2: Divide the Circle into 7 Equal Arcs (Central Angles)

The central angle for each side of a regular heptagon inscribed in a circle is given by \( \theta=\frac{360^{\circ}}{7}\approx51.43^{\circ} \).

• Mark a starting point on the circumference of the circle.
• Using a protractor, measure an angle of approximately \( 51.43^{\circ} \) from the starting point and mark the next point on the circumference.
• Repeat this process 6 more times to get 7 equally - spaced points on the circumference of the circle.

Step 3: Connect the Points

Connect the 7 points in order with straight lines using a ruler. This forms one regular heptagon inscribed in the circle.

Step 4: Draw 3 More Heptagons

• For the second heptagon, choose a different starting point on the circumference of the circle (for example, a point that is not one of the vertices of the first heptagon). Repeat steps 2 and 3.
• For the third heptagon, choose another distinct starting point and repeat the process of dividing the circle into 7 equal arcs and connecting the points.
• For the fourth heptagon, choose a final distinct starting point and repeat the process.

Part 2: Drawing 1 Equilateral Triangle in a Circle of Radius 6 cm

Step 1: Prepare the Compass

Set the compass to a radius of \( r = 6\space\text{cm} \). Draw a circle with this radius.

Step 2: Divide the Circle into 3 Equal Arcs (Central Angles)

The central angle for each side of a regular (equilateral) triangle inscribed in a circle is \( \theta=\frac{360^{\circ}}{3} = 120^{\circ} \).

• Mark a starting point on the circumference of the circle.
• Using a protractor, measure an angle of \( 120^{\circ} \) from the starting point and mark the next point on the circumference.
• Measure another \( 120^{\circ} \) from the second point and mark the third point. The third point should be such that when you measure \( 120^{\circ} \) from it, you get back to the starting point.

Step 3: Connect the Points

Connect the three points with straight lines using a ruler. This forms an equilateral triangle inscribed in the circle of radius 6 cm.

(Note: Since this is a drawing - based task, the final "answer" is the successful completion of the two drawing tasks as described above. If a written confirmation is required, we can say that the heptagons and the equilateral triangle have been drawn according to the given instructions.)

Answer:

Part 1: Drawing 4 Inscribed 7 - Sided Polygons (Heptagons) in a Circle of Radius 5 cm

Step 1: Prepare the Compass

Set the compass to a radius of \( r = 5\space\text{cm} \). Draw a circle with this radius. This circle will be the circum - circle for all the heptagons.

Step 2: Divide the Circle into 7 Equal Arcs (Central Angles)

The central angle for each side of a regular heptagon inscribed in a circle is given by \( \theta=\frac{360^{\circ}}{7}\approx51.43^{\circ} \).

• Mark a starting point on the circumference of the circle.
• Using a protractor, measure an angle of approximately \( 51.43^{\circ} \) from the starting point and mark the next point on the circumference.
• Repeat this process 6 more times to get 7 equally - spaced points on the circumference of the circle.

Step 3: Connect the Points

Connect the 7 points in order with straight lines using a ruler. This forms one regular heptagon inscribed in the circle.

Step 4: Draw 3 More Heptagons

• For the second heptagon, choose a different starting point on the circumference of the circle (for example, a point that is not one of the vertices of the first heptagon). Repeat steps 2 and 3.
• For the third heptagon, choose another distinct starting point and repeat the process of dividing the circle into 7 equal arcs and connecting the points.
• For the fourth heptagon, choose a final distinct starting point and repeat the process.

Part 2: Drawing 1 Equilateral Triangle in a Circle of Radius 6 cm

Step 1: Prepare the Compass

Set the compass to a radius of \( r = 6\space\text{cm} \). Draw a circle with this radius.

Step 2: Divide the Circle into 3 Equal Arcs (Central Angles)

The central angle for each side of a regular (equilateral) triangle inscribed in a circle is \( \theta=\frac{360^{\circ}}{3} = 120^{\circ} \).

• Mark a starting point on the circumference of the circle.
• Using a protractor, measure an angle of \( 120^{\circ} \) from the starting point and mark the next point on the circumference.
• Measure another \( 120^{\circ} \) from the second point and mark the third point. The third point should be such that when you measure \( 120^{\circ} \) from it, you get back to the starting point.

Step 3: Connect the Points

Connect the three points with straight lines using a ruler. This forms an equilateral triangle inscribed in the circle of radius 6 cm.

(Note: Since this is a drawing - based task, the final "answer" is the successful completion of the two drawing tasks as described above. If a written confirmation is required, we can say that the heptagons and the equilateral triangle have been drawn according to the given instructions.)