CK-12 Geometry
Page 35
Find the measure of arcs in a circle.
Introduction
Chords are line segments whose endpoints are both on a circle. The figure shows an arc and its related chord .
There are several theorems that relate to chords of a circle that we will discuss in the following sections.
Perpendicular Bisector of a Chord
Theorem The perpendicular bisector of a chord is a diameter.
Proof
We will draw two chords, and such that is the perpendicular bisector of .
We can see that for any point on chord .
The congruence of the triangles can be proven by the postulate:
and are right angles
This means that .
Any point that is equidistant from and lies along , by the perpendicular bisector theorem. Since the center of the circle is one such point, it must lie along so is a diameter.
If is the midpoint of then and are radii of the circle and is a diameter of the circle.
Perpendicular Bisector of a Chord Bisects Intercepted Arc
Theorem The perpendicular bisector of a chord bisects the arc intercepted by the chord.
Proof
We can see that because of the postulate.
and are right angles.
This means that .
This completes the proof.
Congruent Chords Equidistant from Center
Theorem Congruent chords in the same circle are equidistant from the center of the circle.
First, recall that the definition of distance from a point to a line is the length of the perpendicular segment drawn to the line from the point. and are these distances, and we must prove that they are equal.
Proof.
by the SSS Postulate.
Since the triangles are congruent, their corresponding altitudes and must also be congruent: .
Therefore, and are equidistant from the center.
Converse of Congruent Chords Theorem
Theorem Two chords equidistant from the center of a circle are congruent.
This proof is left as a homework exercise.
Next, we will solve a few examples that apply the theorems we discussed.
Example 1
, and is . from the center of circle .
A. Find the radius of the circle.
B. Find
Draw the radius .
A. is the hypotenuse of the right triangle .
.; .
Apply the Pythagorean Theorem.
B. Extend to intersect the circle at point .
Example 2
Two concentric circles have radii of and A segment tangent to the smaller circle is a chord of the larger circle. What is the length of the segment?
Start by drawing a figure that represents the problem.
is a right triangle because the radius of the smaller circle is perpendicular to the tangent at point .
Apply the Pythagorean Theorem.
Example 3
Find the length of the chord of the circle.
that is given by line .
First draw a graph that represents the problem.
Find the intersection point of the circle and the line by substituting for in the circle equation.
Solve using the quadratic formula.
or
The corresponding values of are
or
Thus, the intersection points are approximately and .
We can find the length of the chord by applying the distance formula:
Example 4
Let and be the positive intercept and the positive intercept, respectively, of the circle . Let and be the positive intercept and the positive intercept, respectively, of the circle .
Verify that the ratio of chords is the same as the ratio of the corresponding diameters.
For the circle , the intercept is found by setting . So .
The intercept is found by setting . So, .
can be found using the distance formula:
For the circle , and .
The ratio of the .
Diameter of circle is .
Diameter of circle is .
The ratio of the diameters is
The ratio of the chords and the ratio of the diameters are the same.
Lesson Summary
In this section we gained more tools to find the length of chords and the measure of arcs. We learned that the perpendicular bisector of a chord is a diameter and that the perpendicular bisector of a chord also bisects the corresponding arc. We found that congruent chords are equidistant from the center, and chords equidistant from the center are congruent.
Review Questions
Find the value of :
Find the measure of .
Two concentric circles have radii of and A segment tangent to the smaller circle is a chord of the larger circle. What is the length of the segment?
Two congruent circles intersect at points and . is a chord to both circles. If the segment connecting the centers of the two circles measures and , how long is the radius?
Find the length of the chord of the circle that is given by line .
Prove Theorem 9-9.
Sketch the circle whose equation is . Using the same system of coordinate axes, graph the line , which should intersect the circle twice—at and at another point in the second quadrant. Find the coordinates of .
Also find the coordinates for a point on the circle above, such that .
The line intersects the circle in two points. Call the third quadrant point and the first quadrant point , and find their coordinates. Let be the point where the line through and the center of the circle intersects the circle again. Show that is a right triangle.
A circular playing field in diameter has a straight path cutting across it. It is from the center of the field to the closest point on this path. How long is the path?
Review Answers
proof
; ; ; ; ;
Inscribed Angles
Learning Objective
Find the measure of inscribed angles and the arcs they intercept
Inscribed Angle, Intercepted Arc
An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle. An inscribed angle is said to intercept an arc of the circle. We will prove shortly that the measure of an inscribed angle is half of the measure of the arc it intercepts.
Notice that the vertex of the inscribed angle can be anywhere on the circumference of the circle--it does not need to be diametrically opposite the intercepted arc.
Measure of Inscribed Angle
The measure of a central angle is twice the measure of the inscribed angle that intercepts the same arc.
Proof.
and both intercept is a central angle and angle is an inscribed angle.
We draw the diameter of the circle through points and , and let and
We see that is isosceles because and are radii of the circle and are therefore congruent.
From this we can conclude that
Similarly, we can conclude that
We use the property that the sum of angles inside a triangle equals to find that:
and .
Then,
and
Therefore
Inscribed Angle Corollaries a-d
The theorem above has several corollaries, which will be left to the student to prove.
a. Inscribed angles intercepting the same arc are congruent
b. Opposite angles of an inscribed quadrilateral are supplementary
c. An angle inscribed in a semicircle is a right angle
d. An inscribed right angle intercepts a semicircle
Here are some examples the make use of the theorems presented in this section.
Example 1
Find the angle marked in the circle.
The is twice the measure of the angle at the circumference because it is a central angle.
Therefore,
This means that
Example 2
Find the angles marked in the circle.
So,
Example 3
Find the angles marked and in the circle.
First we use to find the measure of angle .
Therefore, .
because they are inscribed angles and intercept the same arc .
In .
Lesson Summary
In this section we learned about inscribed angles. We found that an inscribed angle is half the measure of the arc it intercepts. We also learned some corollaries related to inscribed angles and found that if two inscribed angles intercept the same arc, they are congruent.
Review Questions
In and . Find the measure of each angle:
Quadrilateral is inscribed in such that .
Find the measure of each of the following angles:
In the following figure, and .
Find the following measures:
Prove the inscribed angle theorem corollary a.
Prove the inscribed angle theorem corollary b.
Prove the inscribed angle theorem corollary c.
Prove the inscribed angle theorem corollary d.
Find the measure of angle .
Find the measure of the angles and .
Suppose that is a diameter of a circle centered at , and is any other point on the circle. Draw the line through that is parallel to , and let be the point where it meets . Prove that is the midpoint of .
Review Answers
Proof
Proof
Proof
Proof
Hint: , so .
Angles of Chords, Secants, and Tangents
Learning Objective
find the measures of angles formed by chords, secants, and tangents
Measure of Tangent-Chord Angle
Theorem The measure of an angle formed by a chord and a tangent that intersect on the circle equals half the measure of the intercepted arc.
In other words:
and
Proof
Draw the radii of the circle to points and .
is isosceles, therefore
We also know that, because is tangent to the circle.
We obtain
Since is a central angle that corresponds to then, .
This completes the proof.
Example 1
Find the values of and .
First we find angle
Using the Measure of the Tangent Chord Theorem we conclude that:
and
Therefore,
Angles Inside a Circle
Theorem The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measure of their intercepted arcs. In other words, the measure of the angle is the average (mean) of the measures of the intercepted arcs.
In this figure,
Proof
Draw a segment to connect points and .
Example 2
Find .
Angles Outside a Circle
Theorem The measure of an angle formed by two secants drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs.
In other words:
This theorem also applies for an angle formed by two tangents to the circle drawn from a point outside the circle and for an angle formed by a tangent and a secant drawn from a point outside the circle.
Proof
Draw a line to connect points and .
Example 3
Find the measure of angle .
Lesson Summary
In this section we learned about finding the measure of angles formed by chords, secants, and tangents. We looked at the relationship between the arc measure and the angles formed by chords, secants, and tangents.
Review Questions
Find the value of the variable.
Find the measure of the following angles:
Find the measure of the following angles:
Four points on a circle divide it into four arcs, whose sizes are , and , in consecutive order. The four points determine two intersecting chords. Find the sizes of the angles formed by the intersecting chords.
Review Answers
and
Segments of Chords, Secants, and Tangents
Learning Objectives
Find the lengths of segments associated with circles.
In this section we will discuss segments associated with circles and the angles formed by these segments. The figures below give the names of segments associated with circles.
Segments of Chords
Theorem If two chords intersect inside the circle so that one is divided into segments of length a and b and the other into segments of length b and c then the segments of the chords satisfy the following relationship:
This means that the product of the segments of one chord equals the product of segments of the second chord.
Proof
We connect points and and points and to make and .
Therefore, by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
Example 1
Find the value of the variable.
Segments of Secants
Theorem If two secants are drawn from a common point outside a circle and the segments are labeled as below, then the segments of the secants satisfy the following relationship:
This means that the product of the outside segment of one secant and its whole length equals the product of the outside segment of the other secant and its whole length.
Proof
We connect points and and points and to make and .
Therefore, by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
Example 2
Find the value of the variable.
Segments of Secants and Tangents
Theorem If a tangent and a secant are drawn from a point outside the circle then the segments of the secant and the tangent satisfy the following relationship
This means that the product of the outside segment of the secant and its whole length equals the square of the tangent segment.
Proof
We connect points and and points and to make and .
Therefore, by the AA similarity postulate.
In similar triangles the ratios of corresponding sides are equal.
Example 3
Find the value of the variable assuming that it represents the length of a tangent segment.
Lesson Summary
In this section, we learned how to find the lengths of different segments associated with circles: chords, secants, and tangents. We looked at cases in which the segments intersect inside the circle, outside the circle, or where one is tangent to the circle. There are different equations to find the segment lengths, relating to different situations.
Review Questions
Find the value of missing variables in the following figures:
A circle goes through the points and consecutively. The chords and intersect at . Given that and find ?
Suzie found a piece of a broken plate. She places a ruler across two points on the rim, and the length of the chord is found to be The distance from the midpoint of this chord to the nearest point on the rim is found to be Find the diameter of the plate.
Chords and intersect at . Given and find .
Review Answers
or
Chapter 10: Perimeter and Area
Triangles and Parallelograms
Learning Objectives
Understand basic concepts of the meaning of area.
Use formulas to find the area of specific types of polygons.
Introduction
Measurement is not a new topic. You have been measuring things nearly all your life. Sometimes you use standard units (pound, centimeter), sometimes nonstandard units (your pace or arm span). Space is measured according to its dimension.
One-dimensional space: measure the l
ength of a segment on a line.
Two-dimensional space: measure the area that a figure takes up on a plane (flat surface).
Three-dimensional space: measure the volume that a solid object takes up in “space.”
In this lesson, we will focus on basic ideas about area in two-dimensional space. Once these basic ideas are established we’ll look at the area formulas for some of the most familiar two-dimensional figures.