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Wize High School Grade 9 Math Textbook > Circle Geometry

Introduction to Circles

Surface Area & Volume of Composite ShapesPrevious Section
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Vocabulary of Circles


One common shape in geometry is a circle. One reason why it is so common is because it has lots of great properties associated it. Let's first learn about some terms that are associated with circles.

Vocabulary

A circle is the set of all points in a plane that are a fixed distance from a point called the center of a circle. The distance from the center to a point on the circle is called the radius of a circle.
The notation we use for writing a circle has a circle written next to the name of the center point.

◯A\bigcirc A

Segments

The diameter of a circle is the distance across a circle, through the center point. This is exactly twice the value of the radius.

A chord of a circle is a line segment that has its endpoints on the circle.

Lines

A secant line intersects a circle at two points.

A tangent line intersects a circle at exactly one point.

Properties

We say that two circles are congruent if and only if they have the same radius.

All circles are similar to each other. This is because we can translate one circle until they share a common center point, and then scale it until it is the same.

Tangent line segments to a circle are congruent if they share a common end point.

Example
Use the following diagram to and identify the various objects
  • AB‾\overline{AB}
  • AB↔\overleftrightarrow{AB}
  • Point CC
  • CB‾\overline{CB}
ANSWER:
  • AB‾\overline{AB} is a chord
  • AB↔\overleftrightarrow{AB} is a secant line
  • Point CC is the center of the circle
  • CB‾\overline{CB} is the radius of the circle




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Construct a tangent line to a circle


Our goal in this construction is to create a line that is tangent to a circle at a given point.

Using Technology


Begin with circle ◯A\bigcirc A, and a point BB.
  1. Use the midpoint to draw the midpoint of AB‾\overline{AB}. Label the point as CC.
  2. Use the compass tool to draw a circle with radius AC‾\overline{AC}, centered at CC. Label the intersection of the two circles as point DD.
  3. Draw the line DB↔\overleftrightarrow{DB}
From this we now have that line DB↔\overleftrightarrow{DB} is tangent to circle ◯A\bigcirc A.
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Circles and Tangent Segments


While working on her car Kelly notices two wheels attached by a belt. The situation can model with the following diagram. Here the wheels are the circles and the belt is represented by two tangent segments.

The larger wheel has a radius of 5 inches, while the smaller wheel has a radius of 2 inches. The segment between them is 18 inches.

What is the approximate distance between the center of the wheels?

ANSWER:
Because the segments are tangent, we know that they meet the circles at a right angle. We can use this to break up the space between the gears into a rectangle and a right triangle.
We can then solve for the missing hypotenuse of the right triangle to get the distance between the wheels.

d2=182+32d=182+32d≈18.25\begin{aligned} d^2 &= 18^2 + 3^2 \\ d &= \sqrt{18^2 + 3^2} \\ d &\approx 18.25 \end{aligned}

So the center of the wheels are approximately 18.25 inches apart.

Identify the parts

Match the labeled parts of the diagram with the word that best describes them.

A.
chord
B.
center
C.
radius
D.
diameter
E.
tangent line
BD‾\overline{BD}
point CC
CE‾\overline{CE}
AB↔\overleftrightarrow{AB}
BE‾\overline{BE}

Properties of Circles


Use the properties of circles to find the value of x in the diagram.


Circles and Business

Jeremy is creating a logo for their new coffee shop. The logo will be in the shape of a triangle with a circle on the inside.
Given that AC‾=CE‾=24cm\overline{AC} = \overline{CE} = 24\text{cm}, AE‾=16cm\overline{AE} = 16\text{cm}and that all the sides are tangent to the circle, what is the radius of the circle?