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Wize High School Grade 9 Math Textbook > Circle Geometry

Circles and Polygons

Measuring and CirclesPrevious Section
Angles and Segments in CirclesNext Section
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Angles and Circles


Angles should be a familiar concept at this point, so let's see how they connect with circles.

Vocabulary

An inscribed angle is an angle with a vertex on a circle. The sides of the angle make up chords of the circle.

The arc formed from an inscribed angle is known as an intercepted arc. This arc, is said to subtend the angle.

Properties

The measure of an inscribed angle is exactly one-half the measure of its intercepted arc.

m∠BAC=12mBC⏠or2(m∠BAC)=mBC⏠m\angle{BAC} = \frac{1}{2} m \overgroup{BC} \\ \text{or} \\ 2(m \angle{BAC}) = m \overgroup{BC}
If two inscribed angles of a circle intercept the same arc, then the angles must be congruent.

Polygons

Now that we can form angles we can put these with cords to form entire polygons inside of a circle. These are the same polygons you've seen before, but from being inside of a circle we get additional properties.

When a polygon is inside of a circle we say that it is an inscribed polygon.

If a right angle is inscribed in a circle, the hypotenuse will be the diameter of the circle.

Opposite angles of an inscribed quadrilateral are supplementary.

Example
Use the properties of angles and circles to find the measurement of angle ∠BAC\angle{BAC}
ANSWER:
Using the properties we have that

m∠BAC=12(mBC⏠)=12(24)=12\begin{aligned} m \angle{BAC} &= \frac{1}{2} (m \overgroup{BC}) \\ &= \frac{1}{2}(24) \\ &= 12 \end{aligned}
So the measure of the angle is 12 degrees.
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Construct a circle around a triangle

Our goal in this construction is to create a circle around a given triangle. The circle will go through each vertex of the triangle.

Using Technology


Begin with triangle △ABC\triangle ABC
  1. Construct the perpendicular bisector of AB‾\overline{AB} .
  2. Construct the perpendicular bisector of CB‾\overline{CB}. Label the intersection of these two bisectors as point DD.
  3. Use the compass tool to draw a circle with radius DA‾\overline{DA}, centered at DD.
From this we now have a circle around triangle △ABC\triangle ABC.
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Woodworking and Circles



David is working on building a kitchen table. To make the top, he uses a circular piece of wood. Before he can work on attaching the rest of the table to the top, he'll need to find the center of the wood.


How can David use their carpenters square to find the center of the table?
(Note that a carpenters square is an L-shaped ruler for marking out right angles and measuring distances.)

ANSWER:
If he puts the smaller edge of the carpenters square as a chord of the circle, he can draw out a right angle. By connected the two sides it will form an inscribed right triangle.


From the properties of an inscribed right triangle we know that the hypotenuse will be the diameter of the circle. Using the square to measure this distance, he can then find the middle of it. Since this is the diameter, this will mark out the center of the circle.

Architecture and Circles

The round room is a historic meeting room found in Toronto Canada.
Ashley and Jake both decide to visit the room to take some photographs. For one of the photos they end up standing at the edge of the room in two different spots. While there, they take a picture of a painting on the opposite side of the room.


If both are able to capture the painting exactly with their camera, what does this tell us about the viewing angle of each one?

Inscribed Quadrilateral



Find the value of x in the diagram.

Algebra and Circles



Find the value of x and y in the diagram.
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Construct a square inside a circle


Our goal in this construction is to create a square inside of a given circle.

Using Technology


Begin with circle ◯A\bigcirc A
  1. Draw a line through the center of circle ◯A\bigcirc A. Label the points where the circle intersects the line as BB and CC.
  2. Construct the perpendicular bisector to segment BC‾\overline{BC}. Label the intersection of the bisector and the circle as points DD and EE.
  3. Draw the segments BD‾,DC‾,CE‾,\overline{BD}, \overline{DC}, \overline{CE}, and EB‾\overline{EB}.
From this we now have square □BDCE\square BDCE inside of circle ◯A\bigcirc A.
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Construct a regular pentagon inside a circle

Our goal in this construction is to create a regular pentagon inside of a given circle.

Using Technology


Begin with circle ◯A\bigcirc A
  1. Draw a line through the center of circle ◯A\bigcirc A. Label the points where the circle intersects the line as BB and CC.
  2. Construct the perpendicular bisector to segment BC‾\overline{BC}. Label the intersection of the bisector and the circle as points DD and EE.
  3. Use the midpoint tool to draw the midpoint of AC‾\overline{AC}. Label this as point MM.
  4. Use the compass tool to draw a circle with radius MD‾\overline{MD} , centered at MM. Label the intersection of this circle an BC‾\overline{BC} as NN.
  5. Use the compass tool to draw a circle with radius DN‾\overline{DN}, centered at DD. Label the intersection of this circle and the original circle as P1P_1 . This is the first side of the pentagon.
  6. Repeat step 5 for the other sides of the pentagon.
From this we now have pentagon □DP1P2P3P4\square DP_1P_2P_3P_4 inside of circle ◯A\bigcirc A.