CPhill

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UsernameCPhill
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Questions 47
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 #3
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1)Given angle APS = 77 degrees and arc AP = 123 degrees, find the degree measure of arc PS.

 

If angle APS  = 77°, then  arc AS is twice this = 154°

 

So...arc PS  = (360 - 123 - 154)   =   83°   

 

 

2)In the figure below, EF is a diameter of the circle. Arc AE = 35 degrees and arc CF = 80 degrees. What is the measure of angle ABC, in degrees?

 

Angle AFE  =  (1/2)(35°)  = 17.5°

Angle CEF  = (1/2)(80°)  =  40°

 

So ....angle EBF  = angle ABC  =  (180 - 17.5 - 40)   = 122.5° 

 

 

3)In the diagram, angle A = 30 degrees, arc DE = 170 degrees, and arc BC = 110 degrees. Find the measure of arc CE, in degrees.

 

The sum of arcs CE + DB  = (360 - 110 - 170)  = 80”

 

And  angle A  = (1/2) (arc CE - arc DB).....so....

30  = (1/2)(arc CE - arc DB)

60 = CE - DB

 

So

 

CE + DB  =  80

CE - DB   =  60      add these

 

2CE  =  140     divide both sides by 2

arc CE  = 70°

 

 

4)In circle O, AP is a diameter of length 17 and PS is chord of length 8. What is the length of chord AS?

 

AS  will form a leg of a right triangle with AP  the hypotenuse and PS the other leg....so

 

AS  =  √ [ 17^2  - 8^2 ] =  √ [ 289 - 64 ]   =  √ [ 225]  =   15 units

 

 

5)Grogg draws the following figure. It so happens that angle SAP is 174 degrees less than 4 times angle SOP. Find the degree measure of angle SAP.

 

SAP + 174  = 4*SOP

 

But SAP  and SOP  intercept the same arc PS...so they are equal

So...substituting

 

SAP  + 174  = 4*SAP   subtract SAP from both sides

174   = 3*SAP         dvide both sides by 3

58° = SAP

6)In the diagram, minor arc AB : minor arc BC : major arc CA = 1 : 3 : 5. What is angle ABC in degrees?

 

There are  1 + 3 + 5  =  9 equal arcs in the circle....and angle ABC  spans 5/9  of these

 

So  major arc  CA  =  (5/9) (360)  =  (360/9) * 5  =  40 * 5  =  200°

And since angle ABC is an inscribed angle intercepting this arc, it is 1/2 of this  = 100°

 

7)In the figure, if MR = MK, the measure of arc MK is 130 degrees, and measure of arc MQ is 28 degrees, then what is angle RPK, in degrees?

 

Since MR = MK, then arc MR  = arc MK,,,,and the sum of these arcs is 260”

So minor arc RK   =  360 - 260  = 100°

 

So...since angle RMK intercepts this arc it has 1/2 of its measure  = 50°

 

And since arc MQ is 28, then angle QKM  = 14

 

So in triangle PKM angle KPM  =  180 - 14 - 50  = 116°.....and angle RPK is supplemental to this  = 180 - 116  =  64°

 

 

cool cool cool

CPhill 7 hours ago