Q:

What is the measure of DG?In circle D, mGEC is 230 degreesWhat is the measure of GDC?what is the measure of AC?Segment CO is congruent to segment HZwhich congruence statement is true?A. OZ is congruent to COB. CH is congruent to COZC. CH is congruent to HZOD. CO is congruent to HZin circle K, what is the value of x?A. x=30B. x=25C. x=20D. x=15

Accepted Solution

A:
Problem 1)

Minor arc DG is 110 degrees because we double the inscribed angle (DHG) to get 2*55 = 110

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Problem 2)

Central angle GDC is the same measure as arc GFC. The central angle cuts off this arc.

The arcs GEC and GFC both combine to form a full circle. There are no gaps or overlapping portions.

So they must add to 360 degrees

(arc GEC) + (arc GFC) = 360
(230) + (arc GFC) = 360
(230) + (arc GFC)-230 = 360-230
arc GCF = 130

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Problem 3)

Similar to problem 1, we have another inscribed angle. ABC is the inscribed angle that cuts off minor arc AC

So by the inscribed angle theorem
arc AC = 2*(inscribed angle ABC)
3x+9 = 2*(3x-1.5)

Solve for x
3x+9 = 2*(3x-1.5)
3x+9 = 6x-3
9+3 = 6x-3x
12 = 3x
3x = 12
3x/3 = 12/3
x = 4

If x = 4, then
arc AC = 3x+9
arc AC = 3*4+9
arc AC = 21

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Problem 4)

Since we have congruent chords, this means that the subtended arcs are congruent. In this case, the arcs in question are CO and HZ

So arc CO is congruent to arc HZ

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Problem 5)

We have a right triangle due to Thale's theorem

The angles 75 degrees and x degrees are complementary angles. They must add to 90

x+75 = 90
x+75-75 = 90-75
x = 15