Ray C. Jurgensen, Richard G. Brown
ISBN #9780395977279
1st Edition
2,712 Questions
Homework Questions
The chapter on circles explores the intricate relationships between tangents, chords, arcs, and angles. It provides a comprehensive understanding of central and inscribed angles, as well as practical methods for calculating arc lengths and segments. The integration of technology and historical insights, such as the contributions of Gaetana Agnesi, enriches students' appreciation of geometric analysis and its real-world applications.
1
Understand the fundamental properties and terminology of circles including tangents, chords, arcs, and angles.
2
Analyze the relationships between central angles, inscribed angles, chords, and arcs to solve geometric problems.
3
Apply formulas and principles to compute arc lengths, chord segments, and angle measures in circles.
4
Utilize technology exploration methods to model circle geometry and appreciate historical contributions such as those by Gaetana Agnesi.
CONCEPT
DEFINITION
Circle
A set of all points in a plane that are equidistant from a fixed central point.
Chord
A line segment whose endpoints lie on the circle.
Tangent
A line that touches a circle at exactly one point, forming a right angle with the radius at the point of contact.
Arc
A continuous portion of the circumference of a circle.
Central Angle
An angle whose vertex is at the center of the circle and whose sides intersect the circle, thus defining an intercepted arc.
Inscribed Angle
An angle formed by two chords in a circle with the vertex on the circle. Its measure is half that of its intercepted arc.
Segment
A region inside a circle bounded by a chord and the arc subtended by the chord.
Arc Length
The distance measured along the curved line making up the arc of a circle.
Gaetana Agnesi
An influential mathematician known for her contributions to calculus and geometry, whose work has historical significance in the study of curves and circles.
Draw a circle and several parallel chords. What do you think is true of the midpoints of all such chords?
Draw a circle with center $O$ and a line $\overleftarrow{T S}$ tangent to $\odot O$ at $T .$ Draw $\overline{O T}$, and use a protractor to find $m \angle O T S$.
a. Draw a right triangle inscribed in a circle. b. What do you know about the midpoint of the hypotenuse? c. Where is the center of the circle? d. If the legs of the right triangle are 6 and $8,$ find the radius of the circle.
Plane $Z$ passes through the center of sphere $Q$. a. Explain why $Q R=Q S=Q T$ b. Explain why the intersection of the plane and the sphere is a circle. (The intersection of a sphere with any.plane passing through the center of the sphere is called a great circle of the sphere.)
The radii of two concentric circles are $15 \mathrm{cm}$ and $7 \mathrm{cm} .$ A diameter $\overline{A B}$ of the larger circle intersects the smaller circle at $C$ and $D .$ Find two possible values for $A C$
QUESTION
Given an inscribed angle intercepting an arc of 80 degrees, determine the measure of the inscribed angle.
STEP-BY-STEP ANSWER:
Step 1: Recall that an inscribed angle is equal to half the measure of its intercepted arc. Step 2: Calculate half of 80 degrees, which equals 40 degrees. Final Answer: The measure of the inscribed angle is 40 degrees.
What is the angle between a tangent to a circle and the radius drawn to the point of tangency?
Step 1: Recognize that a tangent to a circle forms a right angle with the radius at the point of tangency. Step 2: Therefore, the angle between the tangent and the radius is 90 degrees. Final Answer: The angle is 90 degrees.
Tangent and Radius