Unit 8 Notes by: Hannah Lee
The main focus of Unit 8 is circles. Secants, tangents, arcs, sectors, and chords are all words that you may not be familiar with at this time. However, by the end of this unit, you will become an expert in just that and many more.
Unit 8A
Lesson 1: Central Angle, Arc Length, and Area of a Sector
The first lesson is all about central angles, arc lengths, and area of sectors, as it implies in the title of this section. These all may seem really confusing, especially if you are unfamiliar with these terms, but the concept itself is actually quite simple. Like with any other topic though, you have to understand the individual words in order to understand the entire concept.
Central Angles are angles whose vertex is on the center of the circle and whose legs are radii that intersect the circle at two distinct points. To find the measure of the central angle, it is actually much easier than it appears! The measure of the central angle is the same as the measure of the intercepted arc. The best way to demonstrate this is in a picture.
Central Angles are angles whose vertex is on the center of the circle and whose legs are radii that intersect the circle at two distinct points. To find the measure of the central angle, it is actually much easier than it appears! The measure of the central angle is the same as the measure of the intercepted arc. The best way to demonstrate this is in a picture.
Arc Length is a distance along an arc that is a portion of the circle's circumference. To find the measure of an arc length, you can multiply the circumference of the circle by the degree measure of the central angle over 360°, or you can form a proportion that states the central angle over 360° is equal to the arc length over the circumference. You can use this equation/proportion to find any part of the equation/proportion, such as the central angle measure, the arc length, and the circumference.
A sector of a circle is a portion or region of the circle bounded by two radii. To find the area of a sector, you can multiply the area of the circle by the degree measure of the central angle over 360°, or you can form a proportion that states the central angle over 360° is equal to the sector area over the area of the entire circle. You can use this equation/proportion to find any part of the equation/proportion, such as the central angle measure, the sector area, and the entire area of the circle.
Central angles, arc lengths, and sector areas just scratch the surface of the world of circles, but they provide the foundation and basis that is needed for the following lessons.
Lesson 2: Radian Measure vs. Degrees
In order to better understand the concept of this lesson, it is best to know the definition of radians and degrees and the difference between the two.
Radians and degrees are both units for angle measurement. However, the two are not as alike as some might initially believe. Degrees are probably the term that you are more familiar with. For example, you know that there are 360 degrees in a circle or 180 degrees in a triangle, but you may not exactly know what degrees actually are. Degrees are used to show the size of the angle and the direction. When you are working with degrees, you tend to use decimals rather than fractions. Some may wonder why we would need radians then. Well, according to Purplemath, we need to work with real numbers in math, and technically, degrees are not considered to be numbers. Like how we need to convert percents into decimals, we convert degrees to radians. The actual definition of radians is the angle that is formed by taking the radius of the circle and wrapping it on the edge of the circle. Many times radians are in fractional form and has pi in the numerator.
Now that we got that out of way, we can move to the next step, which is converting degrees to radians and vice versa.
To covert degrees to radians, you multiply the degree measure by π / 180.
To covert radians to degrees, you multiply the radian measure by 180 / π.
Converting degrees to radians and radians to degrees is actually simpler than it looks! Radians and degrees are both important units for measuring angles, and learning how to convert them between the two is a useful skill to have in the realm of math.
Radians and degrees are both units for angle measurement. However, the two are not as alike as some might initially believe. Degrees are probably the term that you are more familiar with. For example, you know that there are 360 degrees in a circle or 180 degrees in a triangle, but you may not exactly know what degrees actually are. Degrees are used to show the size of the angle and the direction. When you are working with degrees, you tend to use decimals rather than fractions. Some may wonder why we would need radians then. Well, according to Purplemath, we need to work with real numbers in math, and technically, degrees are not considered to be numbers. Like how we need to convert percents into decimals, we convert degrees to radians. The actual definition of radians is the angle that is formed by taking the radius of the circle and wrapping it on the edge of the circle. Many times radians are in fractional form and has pi in the numerator.
Now that we got that out of way, we can move to the next step, which is converting degrees to radians and vice versa.
To covert degrees to radians, you multiply the degree measure by π / 180.
To covert radians to degrees, you multiply the radian measure by 180 / π.
Converting degrees to radians and radians to degrees is actually simpler than it looks! Radians and degrees are both important units for measuring angles, and learning how to convert them between the two is a useful skill to have in the realm of math.
Lesson 3: Properties of Tangents
Lesson three is all about tangents, its properties, and the theorems that come along with it. However, in order to understand this lesson, you must know what a tangent is. A tangent is a line or a line segment that intersects a circle at exactly one point.
Tangents have many different properties and theorems that go along with it! These theorems can be applied to a variety of problems, and sometimes, more than one theorem appears in a problem.
The first theorem states that if a line or line segment is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. This might seem a bit complicated when it is worded in that manner. Basically, the theorem is saying that if a line or line segment is tangent to a circle, then the tangent line is perpendicular to the radius at the point of tangency.
The first theorem states that if a line or line segment is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. This might seem a bit complicated when it is worded in that manner. Basically, the theorem is saying that if a line or line segment is tangent to a circle, then the tangent line is perpendicular to the radius at the point of tangency.
The third theorem states that if two line segments are tangent to a circle from the same external point, then the two line segments are congruent to each other. The best way to explain this theorem is the depict it in a picture. The picture that explains this theorem is below this text.
These theorems can be used in many different ways, especially when they are combined into one problem. For example, let's say that two tangents from the same external point intersect the radius of the circle at the point of tangency. In addition to this, let's say that there is a line drawn from the external point to the center.
This then creates two right triangles. However, there's even more! Since you know that the tangents are congruent, the radii are congruent, and right angles are always congruent to other right angles, you just proved that the two triangles formed are congruent through the SAS congruence theorem! This is because two sides, the tangents and the radii, are congruent to each other, and their included angle, which is the right angles, are congruent to each other. This is a great example of how theorems from other units can help with problems in this unit!
Lesson 4: Properties of Chords
After tangents, we dive straight into the world of chords. A chord is a line segment whose endpoints are on the edge of the circle.
Like the previous lesson, lesson four is packed with postulates and theorems. The first of which may seem a bit obvious at first, but this postulate definitely comes in handy throughout the lesson!
The first postulate is known as the Arc Addition Postulate, and it states the measure of the arc formed by two adjacent arcs is the sum of the measure of the two arcs. Sounds pretty simple, right? Chances are that you already knew the idea of this postulate, but you did not know its name.
The first postulate is known as the Arc Addition Postulate, and it states the measure of the arc formed by two adjacent arcs is the sum of the measure of the two arcs. Sounds pretty simple, right? Chances are that you already knew the idea of this postulate, but you did not know its name.
The first theorem that we learned in this unit actually has three parts to it. In a circle or congruent circles, (1) Congruent central angles have congruent chords, (2) Congruent chords have congruent arcs, and (3) Congruent arcs have congruent central angles. Basically, this all connects together into one flow-chart. One thing leads to another, which leads to another. This can be really helpful when trying to find the measure of an arc or central angle. If you know that the chords are congruent and you know the measure of one of the arcs or central angles, you now know what the other arc or central angle measure is!
From there, we go to the second theorem that we go over in this lesson. It states that in a circle, if the radius or diameter of the circle is perpendicular to the chord, then it bisects the chord and the arc. The same is true with its converse, which states that in a circle, the perpendicular bisector of a chord is a radius or diameter. Typically, problems with this theorem usually incorporate the Pythagorean Theorem, showing how we use theorems from lessons in the past.
The next theorem that we learn is one that states if two chords are congruent, then they are equidistant from the center. Like the previous theorem, the converse of this theorem is also true. The converse states if two chords are equidistant from the center, then the chords are congruent to each other. This is especially helpful when multiple theorems are used in one problem. For example, let's say that there are two chords that are equidistant from the center. Then, that also means that their arcs are also congruent as well as their central angles.
Chords have so many applications to them that can be used! It is good to remember all these theorems, even if some may be difficult to remember.
Unit 8B
Lesson 6: Inscribed Angles and Polygons
Inscribed angles play a huge role in this unit. In order to understand this lesson and later lessons, it is best to know what an inscribed angle and intercepted arc is. An inscribed angle is an angle whose vertex is on the edge of the circle and whose sides contain chords of the circle, which basically means that the angle's sides are also chords of the circle. an intercepted arc is an arc whose endpoints are the points in which the inscribed angle intersects the edge of the circle, which are the two points that are not the vertex. Basically, an intercepted arc is the arc inside the inscribed angle.
Like with the tangents and chords from the two prior lessons, inscribed angles have a lot of theorems that go along with it. The first theorem that we go over is the one that is most common in the world of inscribed angle theorems. The first theorem is known as Inscribed Angle Theorem, and it states that the measure of an inscribed angle is half the measure of its intercepted arc. In other words, the measure of the intercepted arc is twice the measure of its inscribed angle.
The next theorem that is taught is that an inscribed angle subtends a semicircle if and only if the angle is a right angle. This may seem really confusing and complicated, but in fact, it is actually very simple! When a chord or an arc subtends an angle, its endpoints lie on the sides of the angle. Therefore, an inscribed angle that subtends a semicircle has endpoints that are the endpoints of the diameter. The best way to show this is in a picture.
The next theorem brings in a polygon inscribed inside a circle. This theorem states that if a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Unlike the previous theorem, this one is fairly straightforward. Basically, it is saying that angles that are not adjacent to each other, making them opposite of each other, add up to 180 degrees.
The last thing we learned is actually a corollary. This corollary states that if inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. To many of you, this corollary seems really obvious, but despite that, it can be used in many situations.
Inscribed angles and intercepted arcs play a fairly big role in this unit of circles. The information from this lesson, Lesson 6, pops up in other lessons to come as well.
Lesson 7: Angles Created Inside and Outside a Circle
There are many different angles that can be formed around a circle! There are angles whose vertices can be on the edge of the circle, inside the circle, and the outside of the circle, and with all of these angles comes different relationships.
The first relationship we learn is when the vertex is on the edge of the circle. This theorem states that if a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. This basically states that if a tangent and secant (or chord) intersect at a point on the edge of the circle, the measure of the angle that is formed is half the measure of its intercepted arc. Another way to put it is that the measure of the intercepted arc is twice the measure of the angle formed.
The first relationship we learn is when the vertex is on the edge of the circle. This theorem states that if a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. This basically states that if a tangent and secant (or chord) intersect at a point on the edge of the circle, the measure of the angle that is formed is half the measure of its intercepted arc. Another way to put it is that the measure of the intercepted arc is twice the measure of the angle formed.
The second relationship is when the vertex is in the inside of the circle. This theorem states that If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. In a way, the measure of each angle formed is the average of the measure of its intercepted arcs because you find the sum of the measure of its intercepted arcs and divide it by two, the number of intercepted arcs the angle has.
The third and final relationship in this lesson is when the vertex is in the exterior of the circle. This theorem states that if a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs. The clear distinction between the relationships when the vertex is in the interior and in the exterior of the circle is that you subtract the intercepted arcs and divide it by two, instead of finding the sum. It is important to know the difference between the two because if not, it is really easy to make simple mistakes.
Knowing the differences between all three of the relationship lessens the risk of accidentally using the wrong theorem for the wrong question. Therefore, it is always good to double check your work and make sure the answer you get makes sense.
Lesson 8: Segment Relationships in a Circle
Similar to how there were quite a few angle relationships in a circle, there are also plenty of segment relationships in a circle. The segment relationships all depend on what kind of segments they are.
The first segment relationship is when there are two chords. Therefore, this theorem is nicknamed the Chord-Chord Product Theorem. The theorem states that if two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. The theorem is actually really easy. If two chords intersect in the interior of a circle, then there are four segments formed. If you multiply the lengths of the segments of the same chords for each chord, then the products should be equal.
The first segment relationship is when there are two chords. Therefore, this theorem is nicknamed the Chord-Chord Product Theorem. The theorem states that if two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. The theorem is actually really easy. If two chords intersect in the interior of a circle, then there are four segments formed. If you multiply the lengths of the segments of the same chords for each chord, then the products should be equal.
The second segment relationship is when there are two secants, making this theorem known as the Secant-Secant Product Theorem. This theorem states that if two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. The easiest way to think of this relationship is that the product of the whole segment multiplied by the part of that segment outside of the circle should be equal to the product of the other whole segment multiplied by the part of this segment outside of the circle.
The third segment relationship is when there is a secant and tangent, nicknaming this theorem as the Secant-Tangent Product Theorem. This theorem states that if a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. The best way to remember this relationship is that the product of the whole secant segment and the part of the segment in the exterior of the circle is equal to the length of the tangent squared.
These three relationships can be used a lot in the realm of geometry, but like with the angle relationships, it is good to know the differences between these three segment relationships so that simple mistakes are not made.
Lesson 9: Equations of Circles
If you go back to when you took Algebra, you may recall how lines had equations. The equation determined its slope and its y-intercept in the coordinate plane. Similar to that, circles also have equations that have many integral parts to the circle on the graph, including its radius and the center of the circle.
The equation of a circle is: (x - h)² + (y - k)² = r² .
At first, this equation may seem like a combination of a bunch of variables. However, each variable plays a part in the equation. Like it stated before, the equation of a circle tells you the center of the circle and the length of the radius.
To find the length radius, all you have to do is find the square root of the number to the right of the equal side if it is squared already.
The center of the circle is (h, k). This means that the 'x' value of the center is equal to the h in the equation, and the 'y' value of the center is equal to the k in the equation. However, when finding the center of the circle from the equation, you have to be careful of the values.
Let's say that an equation for a circle is (x + 3)² + (y - 2)² = 16.
To find the radius, it is easy. You just square root 16, which is equal to 4.
However, finding the center of the circle may prove to be a predicament. First, let's find the x value of the center. You may think that the x value is positive 3 because in the equation, it has a plus sign in front of the 3! This is where mistakes tend to be made. The x value is, in fact, negative 3. Why? In order to explain this, you have to refer back to the standard equation. Notice how in the standard equation of the circle, there is a negative sign in front of the h. Therefore, if the x value of the center of circle is negative, there will be a positive sign. An easy way to prove this is to plug negative 3 back in the standard equation. If you do this, the first part of the equation will look like: (x - (-3))². When you simplify this, it becomes (x + 3)² ! For that reason, the center of that example circle would be (-3, 2).
Here is a way to remember what happens to the circle based on the h and k:
+h --> moves to the left
-h --> moves to the right
+k --> moves down
-k --> moves right
In addition to this, you can also find the domain and range of the circle. To find the domain, all you have to do is take the farthest point of the circle to the left, which is less than or equal to x, which is less than or equal to the farthest point of the circle to the right. To find the range, you find the lowest point, which is less than or equal to x, which is less than or equal to the highest point of the circle. Another way to find the domain is to use the center of the circle and the length of the radius. In order to find the domain, you have to subtract the length of the radius from the x value of the center. This gives you the point farthest to the left. Then, you add the length of the radius to the x value of the center. This gives you the point farthest to the right. You can use a similar method to find the range. For the range, you have to subtract the length of the radius from the y value of the center, thus giving you the lowest point of the circle. Then, you add the length of the radius to the y value of the center. This gives you the highest point of the circle.
The equation of a circle is: (x - h)² + (y - k)² = r² .
At first, this equation may seem like a combination of a bunch of variables. However, each variable plays a part in the equation. Like it stated before, the equation of a circle tells you the center of the circle and the length of the radius.
To find the length radius, all you have to do is find the square root of the number to the right of the equal side if it is squared already.
The center of the circle is (h, k). This means that the 'x' value of the center is equal to the h in the equation, and the 'y' value of the center is equal to the k in the equation. However, when finding the center of the circle from the equation, you have to be careful of the values.
Let's say that an equation for a circle is (x + 3)² + (y - 2)² = 16.
To find the radius, it is easy. You just square root 16, which is equal to 4.
However, finding the center of the circle may prove to be a predicament. First, let's find the x value of the center. You may think that the x value is positive 3 because in the equation, it has a plus sign in front of the 3! This is where mistakes tend to be made. The x value is, in fact, negative 3. Why? In order to explain this, you have to refer back to the standard equation. Notice how in the standard equation of the circle, there is a negative sign in front of the h. Therefore, if the x value of the center of circle is negative, there will be a positive sign. An easy way to prove this is to plug negative 3 back in the standard equation. If you do this, the first part of the equation will look like: (x - (-3))². When you simplify this, it becomes (x + 3)² ! For that reason, the center of that example circle would be (-3, 2).
Here is a way to remember what happens to the circle based on the h and k:
+h --> moves to the left
-h --> moves to the right
+k --> moves down
-k --> moves right
In addition to this, you can also find the domain and range of the circle. To find the domain, all you have to do is take the farthest point of the circle to the left, which is less than or equal to x, which is less than or equal to the farthest point of the circle to the right. To find the range, you find the lowest point, which is less than or equal to x, which is less than or equal to the highest point of the circle. Another way to find the domain is to use the center of the circle and the length of the radius. In order to find the domain, you have to subtract the length of the radius from the x value of the center. This gives you the point farthest to the left. Then, you add the length of the radius to the x value of the center. This gives you the point farthest to the right. You can use a similar method to find the range. For the range, you have to subtract the length of the radius from the y value of the center, thus giving you the lowest point of the circle. Then, you add the length of the radius to the y value of the center. This gives you the highest point of the circle.
Let's tie everything we just learned in lesson nine in a practice problem! For this circle, let's try to find the center, the radius, the domain, the range, and the equation of the circle in the picture above.
Finding the radius is once again simple. All you have to do is count from the center to the farthest point up, down, left, or right. We can then conclude that the radius is equal to 5.
The center is fairly simple because it is given in the picture. According to the plane, the center of the circle is (0, 1).
To find the domain, we can use either method. For this time, let's use the method where we subtract the length of the radius from the x value of the center and then add the length of the radius to the x value of the center.
0 - 5 = -5
0 + 5 = 5
Therefore, the domain of the circle is: -5 ≤ x ≤ 5.
We can then use the same method to find the range, but instead of using the x value of the center, we use the y value of the center.
1 - 5 = -4
1 + 5 = 6
Therefore, the range of the circle is: -4 ≤ y ≤ 6.
With all that information, we can plug in the variables to find the equation. The standard form of the equation is (x - h)² + (y - k)² = r² . (h, k) is the center, which we know is (0, 1). We also know that the radius is equal to 5.
Now, the equation would look like: (x - 0)² + (y - 1)² = 5² . However, we still need to simplify it.
The simplified equation would be the final answer. Therefore, the equation of this circle is. x² + (y - 1)² = 25.
Now that you finished learning about the last lesson, you are now experts in the field of circles in the realm of geometry!
Finding the radius is once again simple. All you have to do is count from the center to the farthest point up, down, left, or right. We can then conclude that the radius is equal to 5.
The center is fairly simple because it is given in the picture. According to the plane, the center of the circle is (0, 1).
To find the domain, we can use either method. For this time, let's use the method where we subtract the length of the radius from the x value of the center and then add the length of the radius to the x value of the center.
0 - 5 = -5
0 + 5 = 5
Therefore, the domain of the circle is: -5 ≤ x ≤ 5.
We can then use the same method to find the range, but instead of using the x value of the center, we use the y value of the center.
1 - 5 = -4
1 + 5 = 6
Therefore, the range of the circle is: -4 ≤ y ≤ 6.
With all that information, we can plug in the variables to find the equation. The standard form of the equation is (x - h)² + (y - k)² = r² . (h, k) is the center, which we know is (0, 1). We also know that the radius is equal to 5.
Now, the equation would look like: (x - 0)² + (y - 1)² = 5² . However, we still need to simplify it.
The simplified equation would be the final answer. Therefore, the equation of this circle is. x² + (y - 1)² = 25.
Now that you finished learning about the last lesson, you are now experts in the field of circles in the realm of geometry!
To find the making connections portion of the notes, click on the button on the top that states "Making Connections." When you click it, it should lead to a Prezi with the connections between all the lessons.