Lesson 10-3

Example 1  Proofs
Given:	Circle M,  	
Prove:	RNM  CAM

Proof:
Statements	Reasons	
1.	Circle M,  	1.	Given	
2.	  	2.	If 2 minor arcs are , corr. chords are .	
3.	      	3.	All radii of a circle are congruent.	
4.	RNM  CAM	4.	SSS	
5.	RNM CAM	5.	CPCTC	

Example 2  Inscribed Polygons
RECREATION  Dennis built a sandbox in the 	
shape of an octagon. The segments drawn in the 
diagram represent the supports he used to frame 
the sandbox. These supports are diameters of the outer 
circle and create eight congruent central angles. 
Determine the measure of in degrees.

Because all the central angles are congruent, the measure of each numbered angle is 360 ÷ 8 = 45.
Since all radii are congruent, each of the small triangles is isosceles.  Let x be the measure of each base angle in the triangle containing .
	
m5 + x + x	= 180	Angle Sum Theorem
	45 + 2x	= 180	Substitution
	2x	= 135	Subtract 45 from each side.
	x	= 67.5	Divide each side by 2.

6

7

5

Thus, the measure of  is 67.5 degrees.
Fill in the grid with the answer 67.5.  Make sure 
the decimal point is lined up correctly and shade 
in the corresponding bubbles.

Example 3  Radius Perpendicular to a Chord
Circle N has a radius of 36.5 cm. Radius  is perpendicular to chord , which is 53 cm long.
a.	If m = 85, find m.	
	 bisects , so m = m.
m = m	Definition of arc bisector
m = (85) or 42.5	m = 85

b.	Find NZ.
	Draw radius . NZG is a right triangle.
	NG	= 36.5 	r = 36.5
	 bisects . 	A radius perpendicular to a chord bisects it.
	ZG	= (FG) 	Definition of segment bisector
	=  (53) or 26.5	FG = 53

	Use the Pythagorean Theorem to find NZ.
	(NZ)2 + (ZG)2	= (NG)2 	Pythagorean Theorem
	(NZ)2 + (26.5)2	= (36.5)2 	ZG = 26.5, NG = 36.5
	(NZ)2 + 702.25	= 1332.25 	Simplify.
	(NZ)2	= 630	Subtract 702.25 from each side.
	NZ	≈ 25.1	Take the square root of each side.

Example 4  Chords Equidistant from Center
Chords  and  are equidistant from the center. If the radius of circle M is 32, 
find FG and BY.
 and  are equidistant from M, so   .	
Draw  and  to form two right triangles.
Use the Pythagorean Theorem.
	(FN)2 + (NM)2	= (FM)2	Pythagorean Theorem
	(FN)2 + (22)2	= (32)2	NM = 22, FM = 32
	(FN)2 + 484	= 1024	Simplify.
	(FN)2	= 540	Subtract 484 from each side.
	FN	≈ 23.2	Take the square root of each side.

FN = (FG), so FG ≈ 2(23.2) or 46.4.
  , so LY also equals 46.4. BY = LY, so BY ≈ (42.4) or 23.2.