Question

explore & reason
using a compass, make a
design using only circles
like the one shown.

a. what instructions can you give to another student so they can make
a copy of your design?

b. make sense and persevere use a ruler to draw straight line segments to
connect points where the circles intersect. are any of the segments that you
drew the same length? if so, why do you think they are? ⓞ mp.1

habits of mind
communicate precisely what mathematical terms or concepts can you use to
describe your design? ⓞ mp.6

Explanation:

Part A

1. First, decide on a radius length for the circles (let's say length \( r \)).
2. Draw the first circle with center \( O \).
3. Then, using the compass set to radius \( r \), place the compass point on the circumference of the first circle (at points that are \( r \) distance apart around the first circle) and draw new circles. For the given design, we can start by drawing one central circle. Then, mark six equally - spaced points on the circumference of the central circle (since the design has six circles around the center? Wait, looking at the diagram, it seems like there is one central circle and six circles around it? Wait, no, the diagram shows a flower - like design with multiple overlapping circles. Let's correct:

• Step 1: Draw a circle with center \( C_1 \) and radius \( r \).
• Step 2: Mark a point \( P_1 \) on the circumference of the circle centered at \( C_1 \). Draw a circle with center \( P_1 \) and radius \( r \).
• Step 3: Mark a point \( P_2 \) on the circumference of the circle centered at \( C_1 \), such that the angle \( \angle P_1C_1P_2 = 60^\circ\) (since the design is symmetric, likely using 60 - degree angles for symmetry). Draw a circle with center \( P_2 \) and radius \( r \).
• Step 4: Repeat step 3 for four more points \( P_3,P_4,P_5,P_6 \) on the circumference of the circle centered at \( C_1 \), each separated by \( 60^\circ\) from the previous point. The overlapping of these circles (the central one and the six around it) will form the given design.

1. When we connect the intersection points of the circles:

• Since all the circles have the same radius (as per the construction in part A, where we used the same compass setting \( r \) for all circles), the segments connecting the intersection points are chords of the circles.
• Consider two intersection points. Let's take two circles, say the central circle and one of the surrounding circles. The distance between their intersection points: Let the radius of each circle be \( r \). The centers of two adjacent surrounding circles and the center of the central circle form an equilateral triangle (because the distance between the center of the central circle \( C \) and the center of a surrounding circle \( P_i \) is \( r \), and the radius of each circle is \( r \)). The intersection points of two circles with radius \( r \) and center - to - center distance \( r \) form a chord. Using the formula for the length of a chord \( l = 2\sqrt{r^{2}-d^{2}/4}\), where \( d \) is the distance between the centers of the two circles. Here, \( d = r \), so \( l = 2\sqrt{r^{2}-r^{2}/4}=2\sqrt{3r^{2}/4}=\sqrt{3}r \). Also, the distance between the intersection points of the central circle and a surrounding circle, and the intersection points of two adjacent surrounding circles: Since the triangles formed by the centers and the intersection points are equilateral (because all sides are equal to \( r \)), the lengths of the segments connecting the intersection points are equal.
• So, when we draw the line segments connecting the intersection points, many of the segments will be the same length. This is because the circles are all congruent (same radius) and are arranged symmetrically (the centers of the surrounding circles are equally spaced around the central circle), so the distances between the intersection points (which are related to the chords of the circles) are equal.

1. Circle: All the figures used in the design are circles, which are defined as the set of all points in a plane at a given distance (radius) from a given point (center).
2. Radius: All the circles in the design have the same radius, which is the distance from the center of each circle to its circumference.
3. Symmetry (Rotational Symmetry): The design has rotational symmetry. The order of rotational symmetry is 6, because if we rotate the design by \( 60^\circ\) ( \( \frac{360^\circ}{6}\)) around the center of the central circle, the design maps onto itself.
4. Intersection Points: These are the points where two or more circles meet.
5. Chord: The line segments we draw in part B are chords of the circles. A chord is a line segment with both endpoints on the circumference of a circle.

Answer:

1. Choose a radius \( r \) for the compass.
2. Draw a circle (central circle) with center \( C \).
3. Mark 6 points \( P_1,P_2,P_3,P_4,P_5,P_6 \) on the circumference of the central circle, such that the angle between the lines joining the center \( C \) to consecutive points (\( \angle P_iCP_{i + 1}\)) is \( 60^\circ\) (where \( P_7=P_1 \)).
4. For each of the points \( P_1 - P_6 \), draw a circle with center at that point and radius \( r \). The overlapping of these 7 circles (1 central + 6 around) will form the design.

Part B