1.  If the diameter of a circle is 58 cm and chord AB is 20 cm from the center, find the length of the chord.

42
The radius is half the diameter or 29.  Use the Pythag theorem to find the length of half of the chord.  Double to find AB.

3.  A square is inscribed in a circle with a  radius of 8.  Find the perimeter of the square.

Use the 45-45-90 triangle to get the side of the square.  Then multiply this answer by 4 to find the perimeter.

4.  If the measure of arc ABC = 270°, and AC=10 cm, Find the length of OA.

Draw in OC.  Since the last arc of the circle must be 90 -- 360-270=90
then the triangle is a 45-45-90.  ( Central angle AOB = arc measure 90.  Radii of the circle are congruent making the triangle isosceles- last two angles must be = and half of the remaining 90 degrees)  Since AC= 10 divide by   which gives you

5.  If AB is a common tangent, AC =9, BD =3  and CD = 12, find the length of AB. 

Draw perpendicular from D to AC forming a rectangle.  Since BD = 3, the opposite side of the rectangle will be congruent leaving 6 out of the 9 units for the length of CE.  We already know that CD is 12.  Use the pythag. theorem to find the length of ED which is also the length of AB since opposite sides of a rectangle are congruent.

6.  If circle O is inscribed in a triangle with side length of  AD =10,  AC =18  and DC = 11 as shown, find the length of AB

8.5
Do a walk around.  Let x = AB  then tangents to a circle are congruent so the 2nd tangent will be x from point A to the circle.  The remaining pieces of the sides are 10-x  and 18-x.  Then make the other pairs of tangents congruent.  DC= 11 and also 10-x+18-x so 11 = 28-2x.  -2x = - 17 so x = 8.5

7.  Given concentric circles with O as a center,  if VZ is a tangent, and VY is a diameter of the larger circle, find WO when XY = 4 and VZ = 32.

30
Draw in BO as a radius to a tangent. This will form a right angle and cut VZ in two congruent halves. Since XY = 4 then VW = 4 (concentric circles radii subtraction) Let WO = x, the BO = x and you can set up a pythag.  The other way to do this is the

8.  AC is a diameter and arc BC is 5 times the measure of arc AB.  Find the measure <BAC

75
Let x = arc AB and 5x = arc BC   x + 5x = 180, 6x=180, x = 30  5(30)= 150 which is BC arc length.  <BAC is inscribed on that arc and inscribed angles are half their arcs so <BAC = 75

9.   If m<A = 84°, m<C =54°, find the m<DBE.

69  The three angles of a triangle add up to 180 so the last < of the triangle = 42,  An angle formed by two tangents to a circle is supplementary to its inner arc so <F and arc DE are supplements.  This makes DE 138.  Since < B is inscribed it is half the intercepted arc of DE so it is 138/2 or 69

10.  If  BA and BC are tangent to the circle and  m<B = 70°, find the m< ADC.

125
If < B = 70 then arc AC must be its supplement.  Two tangent angle and the inner arc are supplements. arc AC = 110.  The rest of the circle is what's left of  360 or 250.  Since ADC is inscribed on that arc it is 1/2 of 250 or 125 degrees.

11.  Given m < AEB = 70°,  m<F=25°,  find the measure of arc AB.

95  Set up a system of  two equations using the rules for a 2 chord angle and a 2 secant angle 2 chord = 1/2(x+y)   2 sec = 1/2 (x-y)  70 = 1/2(x+y),  25 = 1/2 (x-y)  Multiply both equations by 2 and 140 = x+y and 50 = x-y,  Adding the equations together we get. 190 = 2x as the y's drop out.  Divide by 2 and x = 95 which is what we are looking for.

12.  If AE = 8,  BE = 10 and CD = 21, find the length of CE.

16 or  5  Use the 2 chord power theorem.  The product of the segments on one chord = product of the segments on the other chord.  If CD = 21 let x be one part and 21 - x be the other.  The equation reads (8)(10)= x(21-x)  or   Factor into (x-16)(x-5)  set each factor = 0   x=16 or x=5.  Since we are unsure that the figure is drawn to scale both answers must be written down.

13.  Find the radius of circle O, if CD = 6,  DB = 4 and DC is perpendicular to AB.

6.5
Introduce the rest of the other half of CD.  Since a radius is perpendicular to the chord it bisects the chord so D to the other edge of the circle is 6. Let x = AD. use the 2 chord power theorem The product of the segments on one chord = product of the segments on the other chord.  (6)(6)=4x  4x=36,  x=9.  Since AB = 9+4 or 13 and is a diameter of the circle, the radius of the circle is 13/2 or 6.5