Unit 5 – Section 9.1 – 9.3 Notes

Circle Definitions  

 

  

Name  Definition  Object on Drawing 

Center The point in the middle of the circle. All of the points on the circle are equidistant from this point. 

 

Radius The distance from a point on the circle to the center   

Chord A line segment with each endpoint on the circle   

Diameter A chord that passes through the center of the circle   

Tangent Line A line that intersects the circle at exactly one point   

Point of Tangency The point of intersection between the circle and the tangent line 

 

Central Angle An angle whose vertex is the center of the circle   

Inscribed Angle An angle whose vertex is on the circle   

Minor Arc Any unbroken part of the circumference of a circle that goes less than halfway around the circle 

 

Major Arc Any unbroken part of the circumference of a circle that goes more than halfway around the circle 

 

Intercepted Arc A portion of the circumference of the circle located in the interior of an angle  

 

Secant Is a line that intersects a circle in exactly two points Chord vs. Secant? 

 

Example 1: Use the diagram to the left to practice the vocabulary terms. Find examples of each of the vocabulary words.  

 

Special Relationships between Central angle and inscribed angle Use a compass to create a large circle in the space below. Mark the center point before drawing the circle. Create a central angle, measure it with the protractor and record the measure on the diagram. Create an inscribed angle, from the same intercepted arc, and measure it with a protractor. Be prepared to share your results with the class.

Sample of class results: Measure of Central Angle

Measure of Inscribed Angle

Example 2: The degree measure of an arc is the same as the degree measure of its central angle. The “measure” of an arc is NOT the length of the arc. 

a. Name two minor arcs in the diagram.

b. Name a semicircle in the diagram.

c. Find the measure of minor arc

d. Find the measure of arc

e. Find the measure of major arc

f. What is the sum of the measures of arcs that form a circle?

Prove That All Circles Are Similar We will use similar triangles to prove that all circles are similar. Remember that in similar triangles, 

corresponding angles are congruent and corresponding sides are proportional. Let’s create an isosceles 

triangle in each circle with the vertex angle at the center of the circle and prove the triangles are similar. 

 

 

 

 

 

 

 

 

 

 

Inscribed and Circumscribed Circles Inscribed Circle: The largest possible circle that can be drawn on the inside of a polygon. Each side of the 

polygon must be tangent to the circle. If the circle is inscribed, then the polygon is circumscribed. All 

triangles and all regular polygons have inscribed circles. 

 

 

 

 

 

 

 

 

Circumscribed Circle:  The circle which passes through all the vertices of a polygon. If the circle is 

circumscribed, then the polygon is inscribed. All triangles and regular polygons have circumscribed circles. 

 

 

 

 

 

 

 

 

Additional Theorems Involving Arcs: PARALLEL LINES-CONGRUENT ARCS THEOREM Parallel lines intercept congruent arcs on a circle. Add the measure of arc BD to the diagram.

INTERIOR ANGLE of a CIRCLE THEOREM If an angle is formed by two chords or secants that intersect inside the circle, then the measure of the angle is half the sum of the measures of the intercepted arcs

∠

Measure of angle TVR = ? Show work!

EXTERIOR ANGLE of CIRCLE THEOREM If an angle is formed by chords and/or tangents that intersect outside the circle, then: the measure of the angle is half the difference of the measures of the intercepted arcs. ∠

Measure of angle HFG = ? Show work!

TANGENT to a CIRCLE THEOREM A line drawn tangent to a circle is ______________ to a radius to the point of tangency. Measure this angle with a protractor and record your answer.

 

 

 

Problem Examples Solve for each missing value. 1.   What is the  ?            2.  What is the  ∠ ?     

       What is the  ∠ ?                       

 

 

 

 

 

 

 

 

 

 

 

3.   If  ∠ 98°, what is the measure of ∠ ?    

   

 

 

 

 

 

           

 

 

 

4.   ABCD is inscribed in circle O.  50° and  90°.  What is  ∠ ?           

                        

 

 

 

 

 

 

5.   The circle is inscribed in ∆ . What is the  perimeter of ∆ ? 

   

 

 

 

 

 

 

 

 

 

Unit 5 – Section 9.4 , 9.5, 10.1 Notes

Chords, Tangents & Secants

DIAMETER CHORD THEOREM If a circle’s diameter is perpendicular to a chord, then the diameter bisects the chord and bisects the arc determined by the chord. Mark congruent marks on the diagram.

EQUIDISTANT CHORD THEOREM If two chords of the same circle or congruent circles are congruent, then they are equidistant from the center of the circle. Mark congruent marks on the diagram. Write out the converse to the theorem above Would the converse be true also?

CONGRUENT CHORD, CONGRUENT ARC THEOREM If two chords of the same circle or congruent circles are congruent, then their corresponding arcs are congruent. If two arcs of the same circle or congruent circles are congruent, then their corresponding chords are congruent. SEGMENT CHORD THEOREM If two chords in a circle intersect, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the second chord. So ∙ ∙ _______ Given LK = 6 mm, MK = 4 mm, WK = 3 mm, find KV. Show work! Given LM = 28 in, KV = 21 in, KM = 12 in, find WK. Show work!

TANGENT SEGMENT THEOREM If two tangent segments are drawn from the same point on the exterior of a circle, then the tangent segments are __________________.

SECANT SEGMENT THEOREM IF two secant segments intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the second secant segment and its external secant segment. So (secant1)(external of secant1) = (secant2)(external of secant2) (GC)(HC) = (NC)(PC) Given GH = 10 cm, HC = 6 cm, PC = 5, find NP. Show work!

SECANT TANGENT THEOREM If a tangent and a secant intersect in the interior of a circle, then the product of the lengths of the secant segment and its external secant segment is equal to the square of the length of the tangent segment. So of (ZQ)(YQ) = QR2 Given QY = 2, YZ = 6, find QR. Show work!

INSCRIBED RIGHT TRIANGLE-DIAMETER THEOREM If a triangle is inscribed in a circle such that one side of the triangle is a diameter of the circle, then the triangle is a __________________ triangle. Explain/justify the type of triangle by discussing the measure of the intercepted arc.

Sketch a diagram.

INSCRIBED QUADRILATERAL OPPOSITE ANGLES THEOREM If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. Justify by discussing the measures of the intercepted arcs. Given the quadrilateral TPRS:

∠ ∠ ________ ∠ ∠ __________

Examples:

1. If is a tangent segment and is a radius, what is the measure of ∠ ?

2. If and are tangent segments, what is the measure of ∠ ? 3. If is a tangent segment and is a radius. What is the measure of ∠ ? 4. In the figure shown, the quadrilateral is inscribed inside the circle. The arcs shown have the given measures. Find the measure of each inscribed angle. 5. In the figure shown, quadrilateral ABCD is inscribed in the cirlce. If ∠ 57° and ∠ ≅ ∠ , find ∠ and ∠ . 6. Draw a triangle inscribed in the circle through the three given points, ABC. Decide if ∆ is a right triangle. Explain your answer.

Unit 5 – Section 10.2 – 10.3 Notes

Arc Length & Sector Area

ARC LENGTH What is the name for the distance around a circle, or the length of a circle?

Assume a circle has a circumference of 40 cm and the central angle is 90° or ¼ of the circle.

What would be the length of the intercepted arc? Show work.

Supplies Needed: string, ruler, compass and protractor. is the angle from the center of the circle, or the central angle of the circle. CB is a minor arc of

the circle. We are going to find the length of arc CB. The symbol for arc length is s.

1a. Carefully place the string along the arc. Use the ruler to measure the length of the string, or the length of the arc, in cm. Record the arc length on the diagram. Compare your result with your partner. 1b. Measure the central angle in degrees and record the value on the diagram. 1c. Measure the length of the radius in centimeters and record the value on the diagram.

Arc Length formula:

360°∙ 2

Arc Length = Proportion of Circle * Circumference

Return to the circle above and calculate the arc length. Show work.

Compare this value with your length from measuring the string.

How would the formula be different if the central angle was measured in radians?

Example 1: Calculate the length of in each circle. Express your answer in terms of and then rounded to the nearest hundredth (2 decimal places) A. B. Example 2: Circle O with radius 9 inches ∠ 120° A. What fraction of the circle is the region

bounded by , , and ? B. Calculate the area of Circle O. C. Determine the area of the sector.

Example 3: Circle O with a radius r. ∠ A. What fraction of the circle is the sector? B. Calculate the area of Circle T. C. Determine the area of the sector.

T 

 

Sector Area Formula:

°∙ or ∙

Sector Area = Proportion of Circle * Area of Circle Example 4: Calculate the area of each sector. Express your answer in terms of and then rounded to two decimal places.

A. 7 B. 20 Example 5: Two semicircular cuts were taken from the rectangular region shown. Determine the perimeter of the shaded region. Round your answer to the nearest hundredth (2 decimal places). Example 6: A company has a circular card table with a 4-foot diameter. They want to remove a portion to provide a place for the dealer to stand (see the following diagram). How must surface area of the table will be left for those who are sitting at the table? Round your answer to the nearest hundredth.

  

 

 

 

 

 

Example 1: Find the volume of each prism.

A. B. C.

 

 

 

 

 

Example 2: Calculate the volume of the cylinder.

A. B.

Volume of a Cylinder The volume V of a cylinder is V = Bh = _______, where B is the area of a base, h is the height, And r is the radius of a base.

Volume of a Prism The volume V of a prism is V = _____, where B is the area of the base and h is the height.  

Cavalieri’s Principle Which stack of coins has the bigger volume?

Example 4: Find the volume of the oblique cylinder.

A. B.

 

 

 

 

 

 

A cylinder and cone have the same radius and the same height.

Which solid has the larger volume?

How many times bigger is one solid than the other?   

 

 

 

 

 

 

Example 5: Find the volume of each cone.

A.               B.  

 

 

 

Volume of a Cone The volume V of a cone is V = ________ = _________,where B is the area of the base, h is the height, and r is the radius of the base.

                   

                   

 

 

Example 6: Find the volume of each pyramid. A. rectangular pyramid B. square pyramid C. triangular pyramid    

 

 

 

 

 

 

 

 

 

Volume of a Pyramid The volume V of a pyramid is V = _____, where B is the area of the base and h is the height. *Pyramids are named by the shape of their base.

 

 

 

Example 7: Find the volume of the sphere.

A. B.

 

 

 

Applications

Example 8: Carly asked her parents to make a piñata for her birthday party. Her parents decided to make the piñata in the shape of her favorite dessert, an ice cream cone. They stuffed only the cone portion of the piñata.

The height of the cone is 34”

The length of the diameter of the base is 24”.

Calculate the amount of space in the cone that will be filled with goodies.

Example 9: A typical hot-air balloon is about 75 feet tall and about 55 feet in diameter at its widest point. About how many cubic feet of hot air does a typical hot-air balloon hold? Explain how you determined your answer.

Volume of a Sphere The volume V of a sphere is V = ________, where r is the radius of the sphere.

Example 10: A standard sized sheet of paper measures 8.5 inches by 11 inches. Use two standard sized sheets of paper to create two cylinders. One cylinder should have a height of 11 inches and the other cylinder should have a height of 8.5 inches.

A. Carol predicts that the cylinder with a height of 11 inches has a greater volume. Lois predicts that the cylinder with a height of 8.5 inches has a greater volume. Stu predicts that the two cylinders have the same volume. Who do you think is correct? Why?

B. Determine the radius and the height of each cylinder without using a measuring tool. Cylinder #1

Height: Circumference of Base:

Radius:

Cylinder #2 Height: Circumference of Base: Radius:

C. Calculate the volume of each cylinder to prove or disprove your prediction and determine who was

correct. Cylinder #1

Volume:

Cylinder #2 Volume:

Who is correct?

D. Does the radius or the height have a greater impact on the volume? Explain your reasoning.

E. Consider the volume of the cylinder with a height of 8.5 inches. What radius would be required to create a cylinder with a height of 11 inches that has the same volume?