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Circles With and Without Coordinates

Circle Theorems

An angle can be constructed by the lines and line segments that intersect a circle. In this lesson, the angles related to circles and their properties will be explored.

Catch-Up and Review

Here is some recommended reading before getting started with this lesson.

Explore

Investigating Inscribed Angles of a Circle

Consider a circle with a radius of 4 units. An angle whose sides are two chords of the circle is formed as shown. Move the points B and C along the circle so that the angle is a right angle.

Think about the following questions.

  • When the measure of the angle is what can the chord joining B and C be called?
  • Does the arc that the angle intercepts have a special name? If yes, what is it?

Explore

Inscribed and Central Angles of a Circle

On the circle above, construct an angle with a vertex at the center of the circle. The angle being constructed should also cut off the same arc. In other words, construct the corresponding central angle.

Recall that the measure of an arc is the measure of the central angle that includes the arc. Now, move the points and make conjectures that explain the relationship between the measures of angles and arcs.

Discussion

Inscribed Angle of a Circle

In the circles shown previously, the angle formed by two chords BAC is called an inscribed angle.

An inscribed angle is an angle whose vertex lies on the circumference of a circle and whose sides are secant to the circle.

Discussion

Inscribed Angle Theorem

Observe the measures of the angle and the arc intercepted by the angle. Start by moving B so that B, O, and A are collinear. Then, move it once again so that B, O, and C are collinear.

As can be seen, when B, O, and C are collinear, BC becomes the diameter of the circle, and the angle cuts off the semicircle. Furthermore, the measure of is twice the measure of an inscribed angle that intercepts it. This statement can be restated as a theorem.

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

Inscribed Angle

In this figure, the measure of BAC is half the measure of

Proof


An inscribed angle can be created in three different ways depending on where the angle is located in relation to the center of the circle.
Inscribed Angle
Consider Case II. A diameter can be drawn from A through O, dividing the inscribed angle into two.
Inscribed Angle

Let the measures of BAD and DAC be and respectively.

Inscribed Angle

Because the radii of a circle are all congruent, two isosceles triangles can be obtained by drawing OB and OC.

Inscribed Angle

Therefore, by the Isosceles Triangle Theorem, the measures of OBA and OCA will also be and respectively.

Inscribed Angle

By the Triangle Exterior Angle Theorem, it is recognized that and

Inscribed Angle
The measure of an intercepted arc is the same as the measure of its central angle. Therefore, the measures of and are and respectively.
From here, using the Angle Addition Postulate and the Arc Addition Postulate, mBAC and can be written in terms of and
Consequentially, the measure of BAC is half the measure of The proof of Case II has been completed.

By applying similar logic as the procedure above, Case I and Case III can be proven.

Example

Finding the Inscribed Angle of a Circle

In the diagram, the vertex of UVT is on the circle O, and the sides of the angle are chords of the circle. Given the measure of , find the measure of the angle.

Write the answer without the degree symbol.

Hint

The Inscribed Angle Theorem can be used to find the measure of the angle.

Solution

The angle shown in the diagram fits the definition of an inscribed angle. For this reason, the measure of the angle can be found using the Inscribed Angle Theorem. The theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.

The arc intercepted by UVT measures
Therefore, UVT measures

Pop Quiz

Practice Finding Inscribed Angles

Find the measure of the inscribed angle in the circle.

Inscribed Angle

Example

Finding the Central Angle of a Circle

Similarly, given the measure of an inscribed angle, the measure of its corresponding central angle can be found using the Inscribed Angle Theorem. This can be done because the measure of the central angle is the same as the measure of the arc that the central angle cuts off.

In the circle, KLM measures

Find the measure of the corresponding central angle.

Hint

Start by drawing the corresponding central angle.

Solution

Recall that a central angle is an angle whose vertex lies at the center of the circle. Additionally, the inscribed angle and its corresponding central angle intercept the same arc for this example. Therefore, the corresponding central angle is KOM.

The arc is included in KOM. Therefore, the measure of KOM is equal to the measure of Using the Inscribed Angle Theorem, the measure of can be found.
Solve for
Therefore, KOM also measures

Pop Quiz

Practice Finding Central Angles

Given the measure of an inscribed angle, find the measure of its corresponding central angle.

Inscribed Angle

Discussion

Inscribed Angles of a Circle Theorem

Up to now, the relationship between inscribed angles and their corresponding central angles has been discussed. Now the relationship between two inscribed angles that intercept the same arc will be investigated.

As can be observed, the angles are congruent, so long as they intercept the same arc.

If two inscribed angles of a circle intercept the same arc, then they are congruent.

A circle with two inscribed angles that intercept the same arc

By this theorem, ADB and ACB in the above diagram are congruent angles.

ADBACB

Proof

Consider two inscribed angles ADB and ACB that intercept the same arc in a circle.

Two inscribed angles intercept the same arc
By the Inscribed Angle Theorem, the measures of ADB and ACB are half the measure of
Therefore, by the Transitive Property of Equality, it can be said that these two angles have the same measure.
Consequently, by the definition of congruent angles, ADB and ACB are congruent.

ADBACB

Example

Using the Inscribed Angles of a Circle Theorem

Mark and Jordan have been asked to find the measure of G.

Mark claims that the measure of G is the same as the measure of E. Jordan, however, thinks that its measure is half the measure of E. Who is correct?
Find the measure of G.

Hint

Determine which angles intercept the same arc. Use the Inscribed Angles of a Circle Theorem to find mG, .

Solution

The inscribed angles E and H intercept

By the Inscribed Angles of a Circle Theorem, E is congruent to H.
Accordingly, half the measure of E can be found as Additionally, F and G intercept the same arc,
Again by the Inscribed Angles of a Circle Theorem, F is congruent to G. Therefore,
Since neither Mark's claim nor Jordan's claim is correct.

Discussion

Constructing Angles Outside of a Circle

Inscribed angles, or the central angles, are not the only angles related to circles. In the next part, the angles constructed outside the circles will be examined. To construct an angle outside a circle, tangents can be used.

Concept

Tangent

A tangent is a line, segment, or ray that intersects a circle at exactly one point. The point is called point of tangency, and the line, segment, or ray is said to be tangent to the circle.

Tangent and Point of Tangency

Rule

Tangent to a Circle Theorem

A line is a tangent to a circle if and only if the line is perpendicular to the endpoint of a radius on the circle's circumference.

Circle Q and tangent line m

Based on the diagram, the following relation holds true.

Line m is tangent to

Proof

The theorem will be proven in two parts as it is a biconditional statement. Each will be proven by using an indirect proof.

Part I: If a Line Is a Tangent to a Circle, Then It Is Perpendicular to the Endpoint of a Radius on the Circle’s Circumference

Assume that line m is tangent to the circle centered at Q and not perpendicular to QP. By the Perpendicular Postulate, there is another segment from Q that is perpendicular to m. Let that segment be The goal is to prove that QP must be that segment. The following diagram shows the mentioned characteristics.

Line m is perpendicular to segment QT but not perpendicular to segment QP.
Since QT is said to be perpendicular to m, QTP is a right triangle. In this case, QP, whose length is r units, is the hypotenuse and, therefore, the longest side. As a result, QT is less than QP.
There is a part of QT that lies outside of the circle. Let b be the length of that part. The other part of QT is the radius of because it is a segment from the center of the circle to a point on the circle.
Segment QT has a length of b plus r.
By the Segment Addition Postulate, QT is equal to the sum of r and b. This expression, together with QP=r, can be substituted into the aforementioned inequality.
QT<QP
r+b<r
b<0
Since a length cannot be less than 0, the assumption that QP is not perpendicular to the line m must be false. Therefore, QP is perpendicular to the tangent m at its endpoint on the circle. This concludes the first part of the proof.

Line m is tangent to

Part II: If a Line Is Perpendicular to the Endpoint of a Radius on a Circle’s Circumference, Then It Is a Tangent to the Circle

For the second part, it will be assumed that m is perpendicular to the radius QP at P, and that line m is not tangent to In this case, line m intersects at a second point R.

Line m is perpendicular to segment QP but not tangent to the circle.
Since m is said to be perpendicular to QP at P, QPR is a right triangle and QR is the hypotenuse. Therefore, QP is less than QR.
However, it can also be seen that both QP and QR are radii of Therefore, they have the same length.
These two statements are contradictory to each other. Therefore, both cannot be true. The contradiction came from supposing that line m was not tangent to Consequently, m must be a tangent line to the circle. This completes the second part of the proof.

line m is tangent to

Having proven both parts, the proof of the biconditional statement of the theorem is now complete.

Line m is tangent to

The method used to prove the theorem is called indirect proof.

Example

Using the Tangent to Circle Theorem

In the diagram, is tangent to the circle at the point A, and DA is a diameter.

If the measure of ABC is Find the measure of

Hint

A tangent is perpendicular to the radius of a circle through the point of tangency.

Solution

It has been given that By the Tangent to Circle Theorem, is perpendicular to DA. In other words,

Since the sum of the measures of the interior angles of a triangle is
Notice that ACD is an inscribed angle on the diameter AD of Therefore, ACD measures
Once again, using the Triangle Sum Theorem, the measure of DAC can be found to be
Since is associated with DAC and the measure of the arc can be found using the Inscribed Angle Theorem.
Solve for

Discussion

Circumscribed Angle Theorem

When two tangents of a circle intersect at an exterior point, the angle formed is called a circumscribed angle.

In the diagram above, and are tangents to the circle and BAC is a circumscribed angle.

A circumscribed angle is supplementary to the central angle it cuts off.

The measure of a circumscribed angle is equal to minus the measure of the central angle that intercepts the same arc.

Considering the above diagram, the following relation holds true.

Proof

By definition, a circumscribed angle is an angle whose sides are tangents to a circle. Since ADB is a circumscribed angle, and are tangents to at points A and B, respectively. By the Tangent to Circle Theorem, CA is perpendicular to and CB is perpendicular to

Notice that ADBC is a quadrilateral and two of its angles are right angles. Recall that the sum of all of the angles in a quadrilateral is Substituting the known angle measures and solving for mADB will give the desired equation.
Solve for mADB
This completes the proof.

Pop Quiz

Practice Using the Circumscribed Angle Theorem

Find the measure of the central angle.

Inscribed Angle

Example

Using Circle Theorems to Solve Problems

The following example involving circumscribed angles and inscribed angles could require the use of the previously learned theorems.

Two tangents from P to are drawn. The measure of KPL is

Find the measure of the inscribed angle that intercepts the same arc as KPL?

Solution

The inscribed angle that intercepts is KNL. By the Inscribed Angle Theorem, mKNL is half
Recall that the measure of an arc is the same as the measure of its corresponding central angle. Therefore, mKML is equal to
Substitute mKML into the above equation.
Now, by the Circumscribed Angle Theorem, mKML is minus mKPL.
It is given that Substituting that measure into the equation will give the measure of the inscribed angle.
Evaluate right-hand side

Closure

Summarizing Angle Theorems Related to Circles

This lesson defined three angles related to circles as well as the relationships between these angles. The diagram below shows the definitions and the main theorems of this lesson.

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