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Nicholas Cook
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Honors Geometry Homework Answers For Section 9.2



Honors Geometry Homework Answers For Section 9.2




If you are taking Honors Geometry, you might be wondering how to find the answers for section 9.2 of your homework. This section covers tangents to circles, which are lines that touch a circle at exactly one point. In this article, we will review some key concepts and formulas related to tangents, and show you how to solve some sample problems from section 9.2.




Honors Geometry Homework Answers For Section 9.2



What is a Tangent to a Circle?




A tangent to a circle is a line that intersects the circle at exactly one point, called the point of tangency. The tangent line is perpendicular to the radius of the circle at the point of tangency. This means that the angle between the tangent line and the radius is 90 degrees.


Here is an example of a tangent line to a circle:


In this figure, line AB is a tangent to circle O at point P. The radius OP is perpendicular to AB at P.


How to Find the Length of a Tangent Segment?




A tangent segment is a part of a tangent line that lies outside the circle. For example, in the figure above, AP and PB are tangent segments. To find the length of a tangent segment, we can use the Pythagorean Theorem or similar triangles.


The Pythagorean Theorem states that for any right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In other words, if a right triangle has sides a, b, and c, where c is the hypotenuse, then:


a + b = c


We can use this theorem to find the length of a tangent segment if we know the radius of the circle and the distance from the center of the circle to the tangent line. For example, in the figure below, we can find the length of AP using the Pythagorean Theorem:


In this figure, OP is the radius of circle O, and OM is perpendicular to AB at M. We are given that OP = 5 cm and OM = 12 cm. We want to find AP.


We can see that triangle OPM is a right triangle with hypotenuse OP and legs OM and PM. We can use the Pythagorean Theorem to find PM:


OP + PM = OM


5 + PM = 12


25 + PM = 144


PM = 144 - 25


PM = 119


PM = 119 cm 10.9 cm


We can also see that triangle AMP is a right triangle with hypotenuse AP and legs AM and PM. We can use the Pythagorean Theorem again to find AP:


AM + PM = AP


(OM - OP) + PM = AP


(12 - 5) + (119) = AP


(7) + 119 = AP


(49) + 119 = AP


(168) = AP


(AP) = 168 cm 13 cm


The length of AP is about 13 cm.


An Alternative Method Using Similar Triangles




We can also find the length of a tangent segment using similar triangles. Two triangles are similar if they have the same shape but not necessarily the same size. This means that their corresponding angles are equal and their corresponding sides are proportional.


We can use similar triangles to find the length of a tangent segment if we know two radii of the circle and one tangent segment. For example, in the figure below, we can find PB using similar triangles:


In this figure, OP and OQ are radii of circle O, and AB is a tangent line at P and Q. We are given that OP = OQ = 8 cm and AQ = 15 cm. We want to find PB.


We can see that triangle OPB and triangle OQA are similar because they have two pairs of congruent angles: OPB OQA (both are right angles) and POB QOA (both are equal to half of POQ). Therefore, their corresponding sides are proportional:


PB/AQ = OB/OA


We can plug in the given values and solve for PB:


PB/15 = (OB - OP)/(OA - OQ)


PB/15 = (OB - 8)/(OA - 8)


PB = 15(OB - 8)/(OA - 8)


We can simplify this expression by noticing that OB and OA are both equal to half of AB (because AB is bisected by O). Therefore, we can write:


PB = 15(OB - 8)/(OA - 8)


PB = 15(AB/2 - 8)/(AB/2 - 8)


PB = (15/4)(AB - 16)/(AB - 16)


PB = (15/4)AB/(AB - 16)


The length of PB depends on AB, which we don't know yet. However, we can use another pair of similar triangles to find AB: triangle AOP and triangle AOQ. These triangles are similar because they have two pairs of congruent angles: AOP AOQ (both are equal to half of POQ) and OAP OAQ (both are vertical angles). Therefore, their corresponding sides are proportional:


AQ/AP = AO/AO


We can plug in the given values and solve for AP:


AQ/AP = AO/AO


(15)/AP = (AO)/(AO)


(15)/AP = (1)


(AP) = (15)


The length of AP is 15 cm. Now we can use this value to find AB:


(AB) = (AP) + (PB)


(AB) = (15) + (PB)


We can plug in our expression for PB in terms of AB:


(AB) = (15) + ((15/4)AB/(AB - 16))


(AB)(AB - 16) = ((15)(AB - 16)) + ((15/4)(AB))


((4)(AB)(AB - 16))= ((60)(AB - 16)) + ((15)(AB))


((4)(AB))= ((60)(AB)) + ((75)(AB))


((4)(AB)


How to Find the Measure of an Angle Formed by a Tangent and a Chord?




A chord of a circle is a line segment that joins two points on the circle. An angle formed by a tangent and a chord has its vertex on the circle. For example, in the figure below, angle APB is formed by tangent AP and chord PB.


To find the measure of an angle formed by a tangent and a chord, we can use the following theorem:


Theorem: The measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc.


An intercepted arc is the part of the circle that lies between two points on the circle. For example, in the figure above, arc PQ is intercepted by chord PB.


We can use this theorem to find the measure of angle APB if we know the measure of arc PQ. For example, in the figure below, we are given that arc PQ has a measure of 80 degrees. We want to find angle APB.


We can use the theorem to find angle APB:


mAPB = (1/2)m(arc PQ)


mAPB = (1/2)(80)


mAPB = 40


The measure of angle APB is 40 degrees.


How to Find the Measure of an Angle Formed by Two Tangents?




An angle formed by two tangents has its vertex outside the circle. For example, in the figure below, angle AOB is formed by tangents OA and OB.


To find the measure of an angle formed by two tangents, we can use the following theorem:


Theorem: The measure of an angle formed by two tangents is equal to half the difference of the measures of the intercepted arcs.


An intercepted arc is the part of the circle that lies between two points on the circle. For example, in the figure above, arcs AB and CD are intercepted by tangents OA and OB.


We can use this theorem to find the measure of angle AOB if we know the measures of arcs AB and CD. For example, in the figure below, we are given that arc AB has a measure of 120 degrees and arc CD has a measure of 60 degrees. We want to find angle AOB.


We can use the theorem to find angle AOB:


mAOB = (1/2)(m(arc AB) - m(arc CD))


mAOB = (1/2)(120 - 60)


mAOB = (1/2)(60)


mAOB = 30


The measure of angle AOB is 30 degrees.


How to Find the Measure of an Angle Formed by Two Chords?




An angle formed by two chords has its vertex inside the circle. For example, in the figure below, angle APB is formed by chords AB and PB.


To find the measure of an angle formed by two chords, we can use the following theorem:


Theorem: The measure of an angle formed by two chords is equal to half the sum of the measures of the intercepted arcs.


An intercepted arc is the part of the circle that lies between two points on the circle. For example, in the figure above, arcs AB and PQ are intercepted by chords AB and PB.


We can use this theorem to find the measure of angle APB if we know the measures of arcs AB and PQ. For example, in the figure below, we are given that arc AB has a measure of 100 degrees and arc PQ has a measure of 40 degrees. We want to find angle APB.


We can use the theorem to find angle APB:


mAPB = (1/2)(m(arc AB) + m(arc PQ))


mAPB = (1/2)(100 + 40)


mAPB = (1/2)(140)


mAPB = 70


The measure of angle APB is 70 degrees.


How to Find the Measure of an Angle Formed by a Secant and a Tangent?




A secant to a circle is a line that intersects the circle in two points. An angle formed by a secant and a tangent has its vertex outside the circle. For example, in the figure below, angle AOB is formed by secant OA and tangent OB.


To find the measure of an angle formed by a secant and a tangent, we can use the following theorem:


Theorem: The measure of an angle formed by a secant and a tangent is equal to half the difference of the measures of the intercepted arcs.


An intercepted arc is the part of the circle that lies between two points on the circle. For example, in the figure above, arcs AB and CD are intercepted by secant OA and tangent OB.


We can use this theorem to find the measure of angle AOB if we know the measures of arcs AB and CD. For example, in the figure below, we are given that arc AB has a measure of 120 degrees and arc CD has a measure of 80 degrees. We want to find angle AOB.


We can use the theorem to find angle AOB:


mAOB = (1/2)(m(arc AB) - m(arc CD))


mAOB = (1/2)(120 - 80)


mAOB = (1/2)(40)


mAOB = 20


The measure of angle AOB is 20 degrees.


Conclusion




In this article, we have learned how to find the answers for section 9.2 of Honors Geometry homework. We have reviewed the definitions and properties of tangents, chords, secants, and arcs of circles. We have also learned how to use various theorems to find the measures of angles and lengths of segments formed by these lines. We hope this article has helped you understand and solve the problems in section 9.2. For more practice and resources, you can check out the links below:


  • Circle Tangents - Math is Fun



  • Tangent is perpendicular to radius - Khan Academy



  • Geometry Honors 2021-2022 - St. Johns County School District



Thank you for reading and good luck with your Honors Geometry homework! d282676c82


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