Chords of a Circle and Central Angle

Bullitt Central

Geometry

Introduction to Circles

Circles

Key Definitions

A circle is the set of all points in the plane that are the same distance from a specific point, called the center.

Radius is the distance from the center of the circle to its outer rim.

Diameter is a line segment that extends through the center of the circle. It is twice the length of the circle's radius.

A chord is a segment that intersects the circle at two points.

Fill in the Blank

The _________________ of a circle is a line segment that extends through the center of the circle and touches both sides of the circle.

Fill in the Blank

A _________ is a line segment whose endpoints are at the circle edge.

Fill in the Blank

The _________ of a circle is a line segment that extends from the circle center to the rim of the circle. Its length is one-half the length of the circle's diameter.

Fill in the Blank

A circle _________ is a portion of the circle's circumference.

Circles

Key Definitions (continued)

The circumference of a circle is the distance around (the rim of) the circle. The formula for circumference is:

Multiple Choice

The figure to the left represents an oval track.  The rounded portions of the track are semicircles.   What is the length of the track? 

Leave your answer in term of 𝝅.

• 184 + 10 $$\pi$$

• 204 + 5 $$\pi$$

• 184 + 5 $$\pi$$

• 204 + 10 $$\pi$$

Multiple Choice

Find the diameter and radius of a circle with a circumference of 36m.  Round to the nearest hundredth.

• d = 10 m

  r = 5m

• d = 18 m

  r = 9m

• d = 11.46 m

  r = 5.73m

• d = 56.55 m

  r = 28.27m

Circles

Key Definitions (continued)

The area of a circle is region occupied by a circle in a two dimensional plane. The formula for the area of a circle is:

Multiple Choice

The area of a circle is 175 cm². 

What is the circumference? Round to the nearest tenth

• 46.9 cm

• 23.4 cm

• 44.5 cm

• 26.5 cm

Multiple Choice

In the displayed figure, each circle has a radius of 7 inches.  What is the area of the portion outside of the circles but inside the square?  Express your answer in terms of 𝝅.

• 441 - 49𝝅

• 196 - 441𝝅

• 1764 - 441𝝅

• 576 - 144𝝅

Circles

Key Definitions (continued)

A circle angle is an angle whose vertex is the center of a circle.

A circle arc is defined as any portion of the circumference of a circle. It is the distance between any two points traced along the circumference of the circle.

The dimensions (in degrees) of circle angles and the arc which exists between its radii are congruent.

Circles

Key Concepts (continued)

The full circumference around a circle is 360^o.

A major arc is greater than 180^o.

A minor arc is less than 180^o.

A semicircular arc is exactly 180^o (or exactly one-half of a circle).

Multiple Choice

Line segments AB and FE are diameters of circle C.  If ∠FCA≅∠ACD, and the m ∠FCA = 48°, what is the

        measure of arc DB?

• 48^o

• 96^o

• 132^o

• 156^o

Multiple Choice

EC and DB are both diameters of circle O. 

Use the angle measure given to find the measure of arc AB.

• 10^o

• 20^o

• 30^o

• 50^o

Circles

Key Definitions (continued)

The arc length of a circle is the distance along the circumference of a circle where the reference arc exists.

Multiple Choice

Find the arc length of a circle with a central angle of 120° and a diameter of 18 inches.  Round to the nearest tenth.

• 113.1 in

• 56.5 in

• 37.7 in

• 18.8 in

Multiple Choice

An Electrician bent a section of copper wire into a partial circle as shown.  The dimensions are given in feet. What is the length of the section of wire to the nearest hundredth of a foot?

• 1.92 feet

• 3.84 feet     

• 4.80 feet

• 7.68 feet

Circles

Key Definitions (continued)

The portion of a circle enclosed by two radii and the arc adjoing them is defined as the sector of a circle.

The area of a sector of a circle is the amount of space occupied by the boundary of the sector (radii and arc).

Multiple Choice

Find the area of a sector with a central angle of 150° and a diameter of 9.5cm.  Round to the nearest tenth.

• 29.5 cm²

• 118.1 cm²

• 3.1 cm²

• 7.9 cm²

Multiple Choice

A water sprinkler rotates in a circular pattern with a radius of 8 feet.  If the sprinkler stops 60° short of completing the circular pattern, what is the approximate area of the watered region?

• 33.49 ft²

• 41.87 ft²

• 167.47 ft²

• 209.96 ft²

Draw

A scientist enlarged a circle portion of a microscope slide to make it easier to count the number of bacteria present in the sample.  He divided the circle into 5 sectors as shown below and plans to count bacteria only in the shaded sector.  What is the area of the shaded sector to the nearest tenth of a square centimeter?

Circles

Key Definitions (continued)

An inscribed angle is an angle with its vertex on the circle and whose sides are chords.

The intercepted arc is the arc that is the inscribed angle and whose endpoints are on the angle. The vertex of an inscribed angle can be anywhere on the circle as long as its sides intersect the circle to form and intercepted arc.

Circles

Key Concept

The inscribed angle theorem states that the measure of an inscribed angle is one-half the measure of its intercepted arc.

Multiple Choice

The playing field for a new game is displayed.  Goals are located at points A, B, and C along the circular boundary.  The center of the field is 0, with the measurements as given.  What is the measure of ∠ABC?

• 55°

• 60°

• 65°

• 120°   

Multiple Choice

In circle O, line segment BC is a diameter.  What is m∠ABC?

• 40°

• 50°

• 80°

• 100°   

Multiple Choice

Find the measure of ∠BAC.

(figure is not drawn to scale).

• 24.5°

• 49°

• 41°

• 98°   

Circles

Key Concept

Inscribed angles that intercept the same arc are congruent. This is called the Congruent Inscribed Angles Theorem and is shown below.

Multiple Choice

Consider the inscribed angles of the circle. 

What is the measure of angle OPN?

• 34.2^o

• 53.5^o

• 87.7^o

• 53.5^o

Circles

Key Concept

If an inscribed angle intercepts a semicircle, the angle is a right angle.

Multiple Choice

In the diagram, KM is a diameter of the circle.  Which of these angles must be a right angle?

• ∠JML

• ∠MJK

• ∠JKM

• ∠JKL

Circles

Key Concept

Where h and k are the x and y coordinates of the circle center respectively, and r is the circle radius.

Multiple Choice

What is the center of the circle represented by the equation: (x -2)² + (y + 3)² = 25?

• (-2, 3)

• (2, 3)

• (-2, -3)

• (2, -3)

Multiple Choice

What is the equation of the circle with center coordinates of (3, -4), and that passes through the point (7, -2)?

• (x-3)² + (y+4)² = 20

• (x+4)² + (y+3)² = 20

• (x+3)² + (y-4)² = 20

• (x–4)² + (y-3)² = 20

Multiple Choice

An upstart package delivery company can only deliver small packages to a portion of the city represented by the encircled area displayed in the coordinate system to the left. Which equation below accurately represents the displayed service boundary of the new company's current delivery area? Also, what is the radius of the service area?

(Note: Each unit on the graph represents one block.) 

• (x+2)² + (y+3)² = 49

  r = 49 blocks

• (x-3)² + (y-2)² = 98

  r = 7 blocks

• (x+2)² + (y+3)² = 49

  r = 7 blocks

• (x-3)² + (y-2)² = 98

  r = 49 blocks

Multiple Choice

A circle has its center at (1, 2).  It has a radius of 5 units. 

Which point does NOT lie on the circle?

• (4, 6) 

• (-2, -2)

• (4, -2)

• (1, 6)