9.3.3F Properties of a Circle
Know and apply properties of a circle to solve problems and logically justify results.
For example: Show that opposite angles of a quadrilateral inscribed in a circle are supplementary.
Overview
Standard 9.3.3 Essential Understandings
The study of geometric figures might be considered one of the "main topics" in geometry, along with, among others, proportional thinking, reasoning and sense making. The focus of this standard is on geometric figures, with the others listed above also being involved to a great extent.
In this standard, students will work with a transversal that intersects two lines. They will study angles formed by the three lines, and determine whether or not the two intersected lines are parallel. They will see many types of triangles, including "special" triangles such as isosceles, equilateral, right, 30-60-90 and 45-45-90 triangles, focusing on the properties of each of these figures.
Students will study similarity and understand that congruence is a special case of similarity. They will work with scale factors in "real-life" applications of similar figures. They will look at quadrilaterals and put them into a hierarchy, with "special" quadrilaterals being "nested" within others. Also, in this standard, students will study circles and their properties, with special focus on their applications.
All Standard Benchmarks
9.3.3.1
Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results.
9.3.3.2
Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results.
9.3.3.3
Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results.
9.3.3.4
Apply the Pythagorean Theorem and its converse to solve problems and logically justify results.
9.3.3.5
Know and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems and logically justify results.
9.3.3.6
Know and apply properties of congruent and similar figures to solve problems and logically justify results.
9.3.3.7
Use properties of polygons-including quadrilaterals and regular polygons-to define them, classify them, solve problems and logically justify results.
9.3.3.8
Know and apply properties of a circle to solve problems and logically justify results.
Benchmark Group F - Circle Properties
9.3.3
9.3.3.8
Know and apply properties of a circle to solve problems and logically justify results.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Calculate circumference and area of a circle, given its radius;
- Calculate arc measures of a circle, given measure of a central angle;
- Calculate arc measures of a circle, given measure of an angle with vertex inside the circle;
- Calculate arc measures of a circle, given measure of an angle with vertex outside the circle.
Work from previous grades that supports this new learning includes:
- Understand, select, and use units of appropriate size and type to measure angles;
- Select and apply techniques and tools to accurately find angle measures to appropriate levels of precision;
- Develop and use formulas to determine and circumference and area of circles.
NCTM Standards
Geometry
Analyze characteristics andproperties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:
- Analyze properties and determine attributes of two- and three-dimensional objects;
- Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;
- Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;
- Use trigonometric relationships to determine lengths and angle measures.
Use visualization, spatial reasoning, and geometric modeling to solve problems:
- Use geometric models to gain insights into, and answer questions in, other areas of mathematics;
- Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
Common Core State Standards (CCSM)
HS.G-CO (Congruence) Prove geometric theorems.
- HS.G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
- HS.G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
- HS.G-CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Define trigonometric ratios and solve problems involving right triangles.
- HS.G-SRT.7. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Apply trigonometry to general triangles.
- HS.G-SRT.10. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
HS.G-C (Circles) Understand and apply theorems about circles.
- HS.G-C.1. Prove that all circles are similar.
- HS.G-C.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
- HS.G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
HS.F-TF (Trigonometric Functions) Extend the domain of trigonometric functions using the unit circle.
- HS.F-TF.3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6.
Misconceptions
Student Misconceptions and Common Errors
- Students will sometimes mix up the formulas for angle measure when the vertex is inside the circle vs. when the vertex is outside the circle.
Vignette
In the Classroom
This vignette illustrates a learning activity that gives students the opportunity to examine circles, including the area of a circle, angles and intercepted arcs.
Note to teachers: These activities are not meant to be completed in one class period. It will take numerous periods to complete the activities, as they relate to many different concepts - area of a circle, circumference of a circle and angles cutting off arcs in circles.
Teacher: Today we're going to look at circles. You might remember that we started by looking at the definition of a circle. We started off with this wording: A circle is the set of all points that are a given distance (radius) from a given point (center). What happened when we looked at that definition?
Student 1: We found it wasn't correct.
Teacher: Why not?
Student 1: What you're describing is a sphere. We need to add "in a plane" to the definition.
Teacher: Right. Now we're going to look at the area of a circle, and try to come up with a formula to calculate the area.
Note: These diagrams come from http://www.geom.uiuc.edu/~huberty/math5337/groupe/rearrange.html
Let's say we take a circle and divide it into eight congruent sectors. It might look like this:
Now, we'll take those eight sectors and rearrange them, like this:
What does this look like to you?
Student 2: Umm, a circle, cut into eight sectors and rearranged?
Teacher: OK, let's cut it up a bit more, into 32 sectors, like this:
Now, if we rearrange these sectors, it might look like this:
Now what does it look like to you?
Student 3: It looks like it might start to form a parallelogram.
Teacher: Good. How do you calculate the formula for the area of a parallelogram?
Student 3: The area of a parallelogram can be found by multiplying the length of the base by the length of the altitude.
Student 4: I think there's a problem here, though.
Teacher: What's that?
Student 4: Well, the "base" of this "parallelogram" is still slightly bumpy, or curved, and not really a segment.
Teacher: Right. How about if we were to cut it into more and more sectors?
Student 4: Then it would look more and more like a parallelogram.
Teacher: Yes, it would. Unfortunately, we don't have the ability to physically cut this into more sectors, so this approximation will have to suffice. If we consider this to approximate a parallelogram, how would you compare the area of this parallelogram to the area of the original circle?
Student 5: I think if we consider this to be a parallelogram, then the area would be the same as the area of the original circle.
Teacher: OK, so what would you consider to be the "altitude" of the parallelogram?
Student 5: That would be the radius of the circle.
Teacher: Right, and what would you consider to be the "base" of the parallelogram?
Student 5: I think that would be the circumference of the circle.
Student 6: But wait, wouldn't half of the circumference of the circle be one of the bases of the parallelogram, and half the circumference be the other base?
Teacher: Yes, so how could you describe one of the bases of the parallelogram?
Student 6: That would be half the circumference of the circle.
Teacher: Good. So, if the area of a parallelogram is found by the following:
Aparallelogram = (base of the parallelogram)(altitude of the parallelogram),
and we consider the area of the circle to be equal to the area of the parallelogram, then we could substitute these:
Acircle = (half the circumference)(radius)
What's the formula for finding the circumference of a circle?
Student 7: Circumference = 2($\pi$)(radius)
Teacher: So what would be a way to find half the circumference?
Student 7: We could just multiply ($\pi$)(radius).
Teacher: OK. Now plug that into the above formula and tell how you could calculate the area of a circle.
Student 7: Acircle = ($\pi$)(radius)(radius) = ($\pi$)(radius)2
Teacher: Next, let's look at circumference. What do you remember about last week's activity, in which you measured two distances relative to a lot of round things?
Student 8: We had a bunch of round objects, and we measured the distance around them (the circumference), and measured the distance across them through the center (the diameter), and when we divided the two, we got a number a little more than three.
Teacher: And when we debriefed about that activity, what did we find?
Student 8: We found that we should get the number pi, so that pi is equal to the circumference of the circle divided by its diameter.
Teacher: Right. Let's take that equation:
$\pi$ = $\frac{circumference}{diamenter}$
and multiply both sides by the length of the diameter. What would your equation look like then?
Student 9: Then, we'd have (diameter)($\pi$) = circumference
Teacher: That's right, the circumference of a circle is equal to its diameter multiplied by pi.
Let's look at angles relative to a circle. First, we'll measure a central angle, which is an angle with vertex at the center of a circle, and the arc it intercepts. It might look like this:
As I move the center, or one of the points on the circle, what do you notice?
Student 9: It seems that the measure of the central angle is equal to the measure of the arc it intercepts.
Teacher: That's true. Let's look now at an inscribed angle, which has its vertex on a circle and whose sides are chords of the circle. For example:
Again, notice what happens as I move the center or a point on the circle.
Student 3: I think the measure of the inscribed angle is half the measure of the arc it intercepts.
Teacher: Right, and keep that "half" idea in mind as we look at some angles whose vertex is outside the circle, like this secant-secant angle.
Student 5: I didn't see what you were getting at until you put the calculator on the screen and subtracted the smaller arc measure from the larger arc measure, and moved the points around so we could see the measures change. Now I see that the measure of the angle is half the difference of the two intercepted arcs.
Teacher: Right, and that will be true for a tangent-tangent angle and for a secant-tangent angle, as well.
Now focus on the angle with vertex inside the circle, like this one:
Teacher: What are you noticing about the measure of the chord-chord angle?
Student 4: Its measure is half the sum of the measures of the arc it intercepts and the arc made by its vertical angle.
Resources
Teacher Notes
- Emphasize the formulas when working with angles that have vertex inside the circle vs. vertex outside the circle vs. vertex on the circle.
Circle Packing
In this unit, students explore circles. In the first lesson students apply the concepts of area and circumference to explore arrangements for soda cans that lead to a more efficient package. They then experiment with three-dimensional arrangements to discover the effect of gravity on the arrangement of soda cans in the second lesson. The final lesson allows students to examine the more advanced mathematical concept of curvature.
Power of Points
In many curricula, the Power of Points theorem is often taught as three separate theorems: the Chord-Chord Power theorem, the Secant-Secant Power theorem and the Tangent-Secant Power theorem. Using a dynamic geometry applet, students will discover that these three theorems are related applications of the Power of Point theorem. They also use their discoveries to solve numerical problems.
Additional Instructional Resources
National Council of Teachers of Mathematics (NCTM). (2010). Focus in High School Mathematics: Reasoning and Sense Making in Geometry. Reston, VA: National Council of Teachers of Mathematics.
central angle: an angle with vertex at the center of a circle.
inscribed angle: an angle with vertex on a circle.
tangent: a line that intersects a circle in exactly one point.
secant: a line that intersects a circle in two points.
chord: a segment with both endpoints on a circle.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- What other instructional strategies can be used to engage students with properties of triangles?
- How can manipulatives be used to help students visualize these abstract concepts?
- How can the instruction be scaffolded for students?
- What additional scaffolding should be provided to ELL students?
- Do the tasks that have been designed connect to underlying concepts or focus on memorization?
- How can it be determined if students have reached this learning goal?
- How can the lesson be differentiated?
Assessment
1. Segment ME is a diameter in circle D. What type of triangle is $\triangle$MTE? Justify your answer.
2. What is the sum of the measures of the five inscribed angles? Justify your answer.
3. The mean radius of Earth is approximately 6370 km. If you're in an airplane (marked A on the diagram), and you know you're 10 km above Earth, how far are you from the horizon (marked H on the diagram)? Justify your answer.
4. For which of these cylindrical-shaped objects does the circumference of the base most closely approximate the height? Justify your answer.
a. Tuna can
b. Cat food can
c. Tennis ball container
d. Cylindrical vase (use the circumference at the top for your estimating purposes)
DOK Level of all Assessments: 3
Assessment hints/solutions:
1.
Since segment ME is a diameter, arc ME must be a semicircle, with measure 180. Angle T is an inscribed angle, so its measure is half the measure of the arc it intercepts, so it has a measure of 90. This means angle T is a right angle, so that triangle MTE is a right triangle.
2.
The sum of the measures of the angles is 180. The measure of angle G is half the measure of arc OM. The measure of angle E is half the measure of arc MT. The measure of angle O is half the measure of arc TG. The measure of angle M is half the measure of arc GE. The measure of angle T is half the measure of arc EO.
This means that the sum of the measures of the angles is half the sum of the measures of the arcs. The sum of the measures of the arcs is 360, so the sum of the measures of the angles must be 180.
3.
Segment AH is tangent to circle M at point H, so angle H is a right angle. We can use the Pythagorean Theorem to find distance AH. Segment HM is 6370 km, and segment AM is 6380 km, so:
63702 + (AH)2 = 63802
and AH is approximately 357 km.
4. The correct answer is C, a tennis ball container. We want the circumference to be approximately equal to the height of the container. The circumference is found by multiplying pi by the diameter of the container. This means that the circumference is a little more than three times the diameter, so the height should be a little more than three times the diameter. Since three tennis balls fit pretty snugly into the container, this is the shape that best answers the question.
Differentiation
- When using these performance assessment items, split them up into multiple days' lessons, being sure to go through review throughout the process.
- Constantly check for vocabulary understanding.
- Extend the learning activities from this framework to the power theorems.
Parents/Admin
Administrative/Peer Classroom Observation
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Students are: (descriptive list) |
Teachers are: (descriptive list) |
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discussing angle measures relative to circles. |
guiding students in calculating angle measures. |
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making conjectures about angle measures. |
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Parent Resources
- Sophia geometry lessons
This social learning community website includes lessons on multiple topics. Most of these lessons were developed by teachers and reviewed. - Khan Academy geometry lessons
This website offers a library of videos and practice exercises on multiple topics.
- Teacher Tube, YouTube
These websites include multiple uploaded lessons on most school topics.
- Mathematics textbooks
Many textbook publishers have websites with additional resources and tutorials. Check your child's textbook for a weblink.