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Inscribed and central
angles^ Unit
Ancient civilizations used astronomy to predict
abundant hunting, planting, or the arrival of winter.
In the astronomical treatise Almagest , the Greco-
Egyptian mathematician Claudius Ptolemy (second
century) made a mathematical description of the
geocentric system (the planets revolve around the
Earth). One of his contributions to mathematics was a
theorem on cyclic quadrilaterals, in which essential
properties of inscribed angles are used.
A sheet from the astronomical treatise Almagest.
The Trigonometry, which studies the relationship
between the sides and angles of a triangle, was
developed by astronomical studies. During the V and VI
centuries, the Indian mathematicians Varahamihira
and Brahmagupta formulated numerous trigonometric
properties using the semi-chord (a triangle inscribed in
the circle with one side as the circle's diameter).
Furthermore, the cyclic quadrilaterals are based on the
study of the inscribed angles
Semi-chord
The angle inscribed ABC is straight This construction allowed the collection of important relationships.
The contents will be developed by addressing the definition of the theorem of the inscribed
angle, which establishes a relationship with the central angle. Also, study the construction
of tangent lines on the circumference, the definition of semi-inscribed angles, and the
relationship between chords and arcs.
Elements of the circumference
Write the name given to the drawn elements on the following circumference:
Segments. Arc.
The segment that goes from the center to a point in the circumference is called radius.
Any portion of the circumference of a circle is called an arc.
Line.
The segment that goes from one point of the circumference to another and passes through the center is called diameter.
The line that touches the circumference at a point is called tangent.
The segment that goes from one point of the circumference to another is called chord.
The point where the tangent line touches the circumference is called: point of tangency.
The elements of the circumference are: The radius to the point of tagency isperpendicular to the tangent point.
The segments: radius, diameter and chord The lines: tangent The arc of the circumference
Write the name of the elements given fore each circimference:
Respond the following questions:
What is the element that is
1 2 in diameter? What is the name of the longest chord on a circumference?
How is the tangent line and radiues to the poingt of tangency of a circumference? By placing two dots on the circumference. How many arcs are formed?
Inscribed angles, part 1
Demonstrate that ∢BOA = 2∢BPA when the center lies somewhere in the ∆BPA.
The diameter is the chord that passes across the center of the circumference.
In ∆AOP: OP = OA (are radiuses of the circumference).
So, ∢OPA = ∢PAO (equal sides oposse equal angles).
Else ∢BOA = ∢OPA + ∢PAO (∢BOA is the external angle of ∆AOP).
Therefore, ∢BOA = 2∢OPA. As ∢OPA = ∢BPA.
Then, ∢BOA = 2∢BPA.
In the inscribed angles whose side coincides with the diameter of the circumference it is satisfied that the measurement of the central angle subtending the same arc is twice the measurement of the inscribed angle.
Determine the value of x for each case.
As
Therefore,
As
Then,
Therefore,
Determine the value of x for each case.
Inscribed angles, part 2
Show that ∢BOA = 2∢BPA when the center is within ∢BPA.
Draw the diameter for QP.
and (as seen in class 3).
Adding both equalities
Therefore, ∢BOA = 2∢BPA.
In the inscribe angles within the central angle, which subtends the same arc, comply that the central angle measure is twice the measurement of the inscribed angle.
Determine the value of x for each case.
As
Therefore,
As
Then,
Therefore, x
Determine the value of x for each case.
Unit
7
Practice what you learned
Determine the value of x for each case.
Determine the value of x and y according to each case.
Unit
7
Congruent arcs
Compare the measurement of ∢BPA with ∢DPC in the following figure, if C�D = A�B.
The notation A�B, means the portion of the arc between point A and point B.
Construct the angles ∢BOA and ∢DOC.
per hypothesis).
and ∢DOC (as per inscribed angle).
Therefore,
In a circumference, the inscribed angles, which subtend arcs of equal measure, have equal measure.
It is also true that if two inscribed angles are of equal measure, then the arcs that they subtend are also of equal measure.
Determine the value of x and y for each case where C�D = A�B.
As ∢BOA = ∢DOC.
Therefore,
As
Therefore,
So, ∢BOA = ∢DOC.
Then,
Determine the value of x and y for each case. Consider A�B = C�D.
Construction of tangents to a circumference
Given the following circumference and the point marked as P, construct with a ruler and compass the lines that pass through point P and are tangent to the circumference.
Taking the midpoint of the segment PO, denoted by Q.
Draw the circumference with the center Q and radius QO.
Draw dots for A and B where the circumference intersects.
Then, ∢OAP = ∢PBO = 90° (both subtend a 180° arc).
Therefore, the lines PA and PB are tangents to the circumference of center O.
The line perpendicular to the radius at a point in the circumference is the tangent to the circumference.
Using the inscribed angle results, one can construct the lines passing through a point P and tangent to a given circumference following the steps of the solution.
Draw a new circumference and P point outside the circumference, and construct the tangents to the circumference passing through the point P.
Based on the exercises in class, respond: (^) You can apply triangle congruence to justify your Are PA and PB segments, the same? answer.
Why?
Chords and arcs of the circumference
In the following figure A�B = C�D. Compare the length of chords AB and CD.
Draw the radiuses for OA, OB, OC and OD.
(because A�B = C�D). To apply the SAS congruence criterion, two sides and the angle between them must be congruent.
(are radiuses of the circumference).
Then, (as per SAS criterion).
Therefore, AB = CD (per congruence).
In a circumference if the measure of the two arcs is equal, then the measure of the chord that subtends those arcs is equal.
In the following figure AB = CD. Compare the length of A�B and C�D arcs.
Draw the radiuses of OA, OB, OC and OD.
Then, ∆BOA ≅ ∆DOC (per LLL criterion).
Then , ∢BOA = ∢DOC (per congruence).
Therefore, A�B = C�D (the central angle is equal).
Points A, B, C, D, E, and F divide the circumference into six equal arcs. Clasify the figures formed by connecting the points indicated in each statement. Look at the example :
BA = BC (because B�A = B�C). R. ABC is an isosceles triangle.
Unit 7
Parallelism
In the figure A�B = C�D. Determine if the segments AD and BC are parallels or secants.
Draw the chord AC.
Then, ∢BCA = ∢DAC (since A�B = C�D).
Therefore, BC ǁ AD (alternate interior angles are equal).
If having two arcs of equal measure on a circumference, then the chords determined by the beginning of one arc and the end of the other are parallel.
Compare the arcs A�B and C�D from the circumference, if BC ǁ AD.
Draw the chord AC.
(alternate interior angles).
Therefore, A�B = C�D (incribed angle theorem).
This result is reciprocal to the initial exercise.
Determine which of the following statements, are sufficient conditions to four consecutive points A, B, C,
and D; on a circumference. Once connected, there is at least a pair of parallel chords.
Unit 7
Four points on a circumference of a circle
Considering ∢ABC = ∢APC and that both angles share the AC segment. It shows that A, B, C and P are on the same circumference.
Point P has three options; on, in or out of the circumference. Option 1 Option 2 Option 3
In this case:
Therefore, A, B, C and P should stay in the same circumference.
Drawing ∢AP'C, then
Since
Therefore,
Drawing ∢AP'C, then
Since
Therefore,
If two equal angles also share a segment at their openings, then the four points are on the same circumference.
Determine the value x and y.
Since ∢CAB = ∢CDB and both share the CB, then A, B, C, D are on the same circumference.
It must satisfied that ∢ BCA = ∢ BDA , then x = 66°. Moreover, it must meet that ∢ CBD = ∢ CAD , then y = 60°.
Determine the value of x and y.
Practice what you learned
Draw a circumference and a dot on the outside of it. Use a ruler and a compass to draw the tangents across P.
Dots A, B, C, D, E, F, G divide the circumference into seven equal arcs. Classify the figures formed by connecting the dots indicated in each statement.
The following figures A, B, C, D are in the circumference. Respond:
What are the angles ∢EAB and ∢EDC?
What are the angles ∢ABE and ∢ACD? Why?
What are the angles ∆ABE and ∆DCE? Why?
Determine which of the following statements are sufficient conditions to four consecutive points A, B, C, D on a circumference. Once connected, there is at least a pair of parallel chords.
Practice what you learned
Determine the value of x or y , accordingly:
