## Learning Math: Geometry

# Parallel Lines and Circles Part C: Circles

## Session 4, Part C

### In This Part

- Inscribed Angles
- More Circle Constructions

A circle is the set of all points in a plane that are equidistant from a given point in the plane, called the center of the circle. **Note 4**

**Problem C1**

To construct the diameter of a circle, you can’t simply draw a segment that looks like it goes through the center; you need to make sure it’s connected to the center of the circle. Here’s one way: Change the Segment Tool to a Ray Tool (or Line Tool) by clicking and holding on that button until the other options appear; then select the one you want. Draw a ray or line with one point on the circle and another point connected to the circle’s center. (Watch your cursor carefully to make sure it doesn’t just look right, but that it really connects to the center.) If you wish, you can draw a segment between the two points where the ray intersects the circle, then hide the ray.

Construct at least three circles of different sizes. For each circle, complete steps (a)-(e).

a. |
Construct the circle’s diameter. (A circle’s diameter is the segment that passes through the center and has its endpoints on the circle.) Constructing the diameter of a circle creates two semicircles. |

b. |
Construct an inscribed angle in one of the semicircles. (An angle is inscribed in a circle if its vertex is on the circle and its rays intersect the circle. For an angle to be inscribed in a semicircle, the rays must intersect the circle at the endpoints of a diameter.) |

c. |
Measure your inscribed angle ∠VXW. |

d. |
Grab the vertex of your inscribed angle with the Pointer Tool and move it around the circle. How does this affect the measure of the angle? |

e. |
Grab the vertex of your inscribed angle with the Pointer Tool and move it around the circle. How does this affect the measure of the angle? |

You can investigate the angles inscribed in semicircles and quarter-circles using Geometer’s Sketchpad.

First, investigate the angle inscribed in a semicircle using Problem C1, if you have not done so already.

**Problem 2**

To investigate the angle inscribed in a quarter-circle, construct three new circles of different sizes and complete steps (a)-(f).

a. |
Construct the circle’s diameter to create two semicircles. |

b. |
Construct a second diameter in the circle that is perpendicular to the first diameter. This creates four quarter-circles. |

c. |
Construct an inscribed angle in one of the quarter-circles. For an angle to be inscribed in a quarter-circle, the rays must intersect the circle at one of the endpoints of each of the diameters. |

d. |
Measure your inscribed ∠ACB. |

e. |
What conjecture can you make about the measure of an angle that is inscribed in a semicircle? |

f. |
What conjecture can you make about the measure of an angle that is inscribed in a quarter-circle? |

### More Circle Constructions

The definition of a circle guarantees that the measure of each radius (the distance from the circle’s center to a point on the circle) is constant for a given circle. So if we want segments that are the same length, we can always rely on the radii of a circle to construct them.

AP, BP, CP, and DP are all radii of circle P; therefore, they all have the same length.

**Problem C3
**Follow the directions below:

a. |
Using the Circle Tool, draw a circle of any size. |

b. |
Construct another circle that has its center on the first circle and that intersects the first circle’s center. |

c. |
Draw segments between the centers of the circles and from each center to one of the points where the circles intersect. |

d. |
What kind of triangle have you formed? How do you know? |

**Problem C4**

Use Geometer’s Sketchpad to create a triangle that is isosceles but not equilateral. How did you do it?

**Problem C5**

A rhombus is a quadrilateral with all four sides equal. Use Geometer’s Sketchpad to construct a rhombus. How did you do it?

### Notes

**Note 4**

Discuss or reflect on the similarities and differences between this definition and how you think of a circle. What are the similarities to the definition of parallel lines? Why is “in the plane” important here, too? Other interesting questions you may want to contemplate: What is the set of all the points in space equidistant from a given point? (a sphere) What is the set of all points in a plane equidistant from a given point not in the plane? (a circle, and the center of the circle is the projection of that point onto the plane along a perpendicular line)

### Solutions

**Problem C1**

The angle has measure 90°. As the vertex moves, the inscribed angle does not change in size. You could make the conjecture that an angle inscribed in a semicircle has measure of 90°.

**Problem C2**

The angle has measure 135°. As the vertex moves, the inscribed angle does not change in size. You could make the conjecture that an angle inscribed in a quarter-circle has measure of 135°.

**Problem C3**

a. |
Answers will vary. |

b. |
Answers will vary. This was the construction from Problem A2. |

c. |
Answers will vary. You should form a triangle by drawing these three segments. |

d. |
The two circles have the same radius. The radius’s length equals the distance between the two centers. Since the distance between a circle’s center and any point on it, including the point where the two circles intersect, is the same as the length of the radius, the base of the triangle (the segment connecting the centers) is the same length as its two other sides. Therefore, the triangle is equilateral. |

**Problem C4**

Answers will vary. One way to construct an isosceles triangle is to start with two circles as in Problem C3. Then connect the two intersection points to each other. Choose the center of one of the circles, and connect it to each of the intersection points. The two radii of the circle are congruent. The angle between them is more than 60° (measure it!), so the third side must be longer than the other two. The triangle is isosceles but not equilateral.

**Problem C5**

Start with the construction of two circles that pass through each other’s centers. Then connect the following points (with segments in between them) in the following order: first intersection point, center of first circle, second intersection point, center of second circle, and back to first intersection point.

All segments are radii of one circle or the other. Since the circles are the same size, all the radii are the same, so the sides of the quadrilateral are all congruent. It must be a rhombus.