CK-12 Geometry
Page 33
What is the measure of in the triangle below?
What is the measure of in the triangle below?
Review Answers
Acute and Obtuse Triangles
Learning Objectives
Identify and use the Law of Sines.
Identify and use the Law of Cosines.
Introduction
Trigonometry is most commonly learned on right triangles, but the ratios can have uses for other types of triangles, too. This lesson focuses on how you can apply sine and cosine ratios to angles in acute or obtuse triangles. Remember that in an acute triangle, all angles measure less than . In an obtuse triangle, there will be one angle that has a measure that is greater than .
The Law of Sines
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of the angle opposite it will be constant. That is, the ratio is the same for all three angles and their opposite sides. Thus, if you find the ratio, you can use it to find missing angle measure and side lengths.
Note the convention that denotes and is the length of the side opposite .
Example 1
Examine the triangle in the following diagram.
What is the length of the side labeled j?
You can use the law of sines to solve this problem. Because you have one side and the angle opposite, you can find the constant that applies to the entire triangle. This ratio will be equal to the proportion of side and . You can use your calculator to find the value of the sines.
So, using the law of sines, the length of is about .
Example 2
Examine the triangle in the following diagram.
What is the measure of ?
You can use the law of sines to solve this problem. Because you have one side and the angle opposite, you can find the constant that applies to the entire triangle. This ratio will be equal to the proportion of side and angle . You can use your calculator to find the value of the sines.
So, using the law of sines, the angle labeled must measure about .
The Law of Cosines
There is another law that works on acute and obtuse triangles in addition to right triangles. The Law of Cosines uses the cosine ratio to identify either lengths of sides or missing angles. To use the law of cosines, you must have either the measures of all three sides, or the measure of two sides and the measure of the included angle.
It doesn’t matter how you assign the variables to the three sides of the triangle, but the angle must be opposite side .
Example 3
Examine the triangle in the following diagram.
What is the measure of side ?
Use the Law of Cosines to find . Since is opposite , we will call the length of by the letter .
So, is about .
Example 4
Examine the triangle in the following diagram.
What is the measure of
Use the Law of Cosines to find the measure of .
So, is about .
Lesson Summary
In this lesson, we explored how to work with different radical expressions both in theory and in practical situations. Specifically, we have learned:
how to identify and use the law of sines.
how to identify and use the law of cosines.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Review Questions
Exercises 1 and 2 use the triangle in the following diagram.
What is the length of side
What is
Examine the triangle in the following diagram.
What is the measure of
Examine the triangle in the following diagram.
What is the measure of
Examine the triangle in the following diagram.
What is the measure of side
Examine the triangle in the following diagram.
What is the measure of
Use the triangle in the following diagram for exercises 7 and 8.
What is the measure of
What is the measure of
Examine the triangle in the following diagram.
What is the measure of
Examine the triangle in the following diagram.
What is the measure of
Review Answers
Chapter 9: Circles
About Circles
Learning Objectives
Distinguish between radius, diameter, chord, tangent, and secant of a circle.
Find relationships between congruent and similar circles.
Examine inscribed and circumscribed polygons.
Write the equation of a circle.
Circle, Center, Radius
A circle is defined as the set of all points that are the same distance away from a specific point called the center of the circle. Note that the circle consists of only the curve but not of the area inside the curve. The distance from the center to the circle is called the radius of the circle.
We often label the center with a capital letter and we refer to the circle by that letter. For example, the circle below is called circle or
Congruent Circles
Two circles are congruent if they have the same radius, regardless of where their centers are located. For example, all circles of radius of are congruent to each other. Similarly, all circles with a radius of are congruent to each other. If circles are not congruent, then they are similar with the similarity ratio given by the ratio of their radii.
Example 1
Determine which circles are congruent and which circles are similar. For similar circles find the similarity ratio.
and are congruent since they both have a radius of .
and are similar with similarity ratio of .
and are similar with similarity ratio of .
and are similar with similarity ratio of .
and are similar with similarity ratio of .
and are similar with similarity ratio of .
Chord, Diameter, Secant
A chord is defined as a line segment starting at one point on the circle and ending at another point on the circle.
A chord that goes through the center of the circle is called the diameter of the circle. Notice that the diameter is twice as long as the radius of the circle.
A secant is a line that cuts through the circle and continues infinitely in both directions.
Point of Tangency and Tangent
A tangent line is defined as a line that touches the circle at exactly one point. This point is called the point of tangency.
Example 2
Identify the following as a secant, chord, diameter, radius, or tangent:
A.
B.
C.
D.
E.
F.
A. is a diameter of the circle.
B. is a radius of the circle.
C. is a chord of the circle.
D. is a tangent of the circle.
E. is a secant of the circle.
F. is a radius of the circle.
Inscribed and Circumscribed Polygons
A convex polygon whose vertices all touch a circle is said to be an inscribed polygon. A convex polygon whose sides all touch a circle is said to be a circumscribed polygon. The figures below show examples of inscribed and circumscribed polygons.
Equations and Graphs of Circles
A circle is defined as the set of all points that are the same distance from a single point called the center. This definition can be used to find an equation of a circle in the coordinate plane.
Let’s consider the circle shown below. As you can see, this circle has its center at point and it has a radius of .
All the points on the circle are a distance of away from the center of the circle.
We can express this information as an equation with the help of the Pythagorean Theorem. The right triangle shown in the fi
gure has legs of length and and hypotenuse of length . We write:
We can generalize this equation for a circle with center at point and radius .
Example 3
Find the center and radius of the following circles:
A.
B.
A. We rewrite the equation as: The center of the circle is at point and the radius is .
B. We rewrite the equation as: The center of the circle is at point and the radius is .
Example 4
Graph the following circles:
A.
B.
In order to graph a circle, we first graph the center point and then draw points that are the length of the radius away from the center.
A. We rewrite the equation as: The center of the circle is point at and the radius is .
B. We rewrite the equation as: The center of the circle is point at and the radius is .
Example 5
Write the equation of the circle in the graph.
From the graph we can see that the center of the circle is at point and the radius is long.
Thus the equation is:
Example 6
Determine if the point is on the circle given by the equation:
In order to find the answer, we simply plug the point into the equation of the circle.
The point satisfies the equation of the circle.
Example 7
Find the equation of the circle whose diameter extends from point to
The general equation of a circle is:
In order to write the equation of the circle in this example, we need to find the center of the circle and the radius of the circle.
Let’s graph the two points on the coordinate plane.
We see that the center of the circle must be in the middle of the diameter.
In other words, the center point is midway between the two points and . To get from point to point , we must travel to the right and up. To get halfway from point to point , we must travel to the right and up. This means the center of the circle is at point or .
We find the length of the radius using the Pythagorean Theorem:
Thus, the equation of the circle is:
Completing the Square:
You saw that the equation of a circle with center at point and radius is given by:
This is called the standard form of the circle equation. The standard form is very useful because it tells us right away what the center and the radius of the circle is.
If the equation of the circle is not in standard form, we use the method of completing the square to rewrite the equation in the standard form.
Example 8
Find the center and radius of the following circle and sketch a graph of the circle.
To find the center and radius of the circle we need to rewrite the equation in standard form. The standard equation has two perfect square factors one for the terms and one for the terms. We need to complete the square for the terms and the terms separately.
To complete the squares we need to find which constants allow us to factors each trinomial into a perfect square. To complete the square for the terms we need to add a constant of on both sides.
To complete the square for the terms we need to add a constant of on both sides.
We can factor the separate trinomials and obtain:
This simplifies as:
You can see now that the center of the circle is at point and the radius is .
Concentric Circles
Concentric circles are circles of different radii that share the same center point.
Example 9
Write the equations of the concentric circles shown in the graph.
Example 10
Determine if the circles given by the equations and are concentric.
To find the answer to this question, we must rewrite the equations of the circles in standard form and find the center point of each circle.
To rewrite in standard form, we complete the square on the and terms separately.
First circle:
The center of the first circle is also at point so the circles are concentric.
Second circle:
The center of the second circle is at point .
Lesson Summary
In this section we discussed many terms associated with circles and looked at inscribed and circumscribed polygons. We also covered graphing circles on the coordinate grid and finding the equation of a circle. We found that sometimes we need to use the technique of completing the square to find the equation of a circle.
Review Questions
Identify each of the following as a diameter, a chord, a radius, a tangent, or a secant line.
Determine which of the following circles are congruent and which are similar. For circles that are similar give the similarity ratio.
For exercises 3-8, find the center and the radius of the circles:
Check that the point is on the circle given by the equation
Check that the point is on the circle given by the equation
Write the equation of the circle with center at and radius .
Write the equation of the circle with center at and radius .
Write the equation of the circle with center at and radius .
For 14 and 15, write the equation of the circles.
In a circle with center one endpoint of a diameter is . Find the other endpoint of the diameter.
The endpoints of the diameter of a circle are given by the points and . Find the equation of the circle.
A circle has center and contains point . Find the equation of the circle.
A circle has center and contains point . Find the equation of the circle.
Find the center and the radius of the following circle: .
Find the center and the radius of the following circle: .
Find the center and the radius of the following circle: .
Determine if the circles given by the equations are concentric. and
and
and
and
Review Answers
is a radius.
is a diameter.
is a tangent.
is a secant.
is a chord.
is a radius.
is congruent to ; is similar to with similarity ratio is similar to with similarity ratio .
The center is located at .
The center is located at .
The center is located at .
The center is located at .
The center is located at .
The center is located at .
The point is on the circle.
The point is not on the circle.
The center is located at .
The center is located at .
The center is located at .
Yes
No
No
Yes
Tangent Lines
Learning Objectives
Find the relationship between a radius and a tangent to a circle.
Find the relationship between two tangents drawn from the same point.
Circumscribe a circle.
Find equations of concentric circles.
Introduction
In this section we will discuss several theorems about tangent lines to circles and the applications of these theorems to geometry problems. Remember that a tangent to a circle is a line that intersects the circle at exactly one point and that this intersection point is called the point of tangency.
Tangent to a Circle
Tangent to a Circle Theorem:
A tangent line is always at right angles to the radius of the circle at the point of tangency.
Proof. We will prove this theorem by contradiction.
We start by making a drawing. is a radius of the circle. is the center of the circle and is the point of intersection between the radius and the tangent line.
Assume that the tangent line is not perpendicular to the radius.
There must be another
point on the tangent line such that is perpendicular to the tangent line. Therefore, in the right triangle is the hypotenuse and is a leg of the triangle. However, this is not possible because . (Note that ).
Since our assumption led us to a contradiction, this means that our assumption was incorrect. Therefore, the tangent line must be perpendicular to the radius of the circle.
Since the tangent to a circle and the radius of the circle make a right angle with each other, we can often use the Pythagorean Theorem in order to find the length of missing line segments.
Example 1
In the figure, is tangent to the circle. Find .
Since is tangent to the circle, then .
This means that is a right triangle and we can apply the Pythagorean Theorem to find the length of .
Example 2
Mark is standing at the top of Mt. Whitney, which is tall. The radius of the Earth is approximately (There are in one mile.) How far can Mark see to the horizon?