4UVEZ4USBUFHZ %XERCISE %NCOURAGE STUDENTS TO GET IN THE HABIT OF DRAWING AND LABELING THE ENTIRE SECANT AS WELL AS THE EXTERNAL SECANT SEGMENT AS SHOWN IN THIS EXERCISE 4HIS WILL HELP THEM AVOID ERRORS IN SETTING UP THE SEGMENT LENGTH EQUATIONS
!"*-$*+,'&%'-&"&*$&$$/ *&
!+"%,*' ,(++ ,$'& $+,'/*,!&,* ',!%'-&"&,!(*(&"-$*"+,& *'%,!&',! (++ ,'",!*+"',!%'-&
5FBDIJOH4USBUFHZ %XERCISE 0OINT OUT THAT AFTER !$ zAND ] !# z DRAWING SEGMENTS ] STUDENTS CAN SHOW THAT Ng%!$ Ng!#$ Ng%#! WHICH WILL GIVE
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
THEM THE PROPORTION ]z %! ]z %# %$
*",,/''$-%&(*'''!'*%
%!
++"%"$*,*"& $++'-,$"&"&,!$&'**'''&(
.BUIFNBUJDBM3FBTPOJOH
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
%XERCISE 3INCE Ng!"# Ng&%# FROM PART B IT CAN BE SHOWN THAT Ng!"# IS A RIGHT TRIANGLE 4HEREFORE STUDENTS CAN USE THE 0YTHAGOREAN 4HEOREM TO FIND THE LENGTH OF ] "# z ] 4HEN THE LENGTH OF #% zCAN BE FOUND USING THE 3EGMENTS OF A #HORD 4HEOREM AND A DIFFERENT PROPOR TION CAN BE USED TO FIND Y WITHOUT SOLVING A SYSTEM OF EQUATIONS
&,!" *%',!/,*/$$ &
*#&'/&*",&)-,"'&'* -+"& ,!+ ,!*%+-*%&,+
,!+("$+/!&,!+&,+ %&,'&,"&+,! &,*',!"*$
'-*+" &"& &&"%,$' ' '*0'-*/+",(*#$+$.('"&, &%'.,',! "*$$'& ,!+ %&,++!'/&+',!,$$',!+(*#$+ *!,!"*$,,!+%,"%(*#$+,*.$*'%('"&, ,'('"&, ,&,"%,*+(*+'&'/+,+!'-$ +(*#$+%'.*'%('"&, ,'('"&,
4WO INTERSECTING CHORDS IN THE SAME CIRCLE 'IVEN $RAW ] !# z AND ] "$ 4WO POINTS z DETERMINE A LINE !#$ !"$ #!" #$" )F TWO INSCRIBED ANGLES OF A CIRCLE INTERCEPT THE SAME ARC THEN THE ANGLES ARE CONGRUENT N !#% N $%" !! 3IMILARITY 0OSTULATE
CM
CM
# $ CM
. +,!($&,'(*'.!'*%
& *+&,+ %&,+
*/ & !'/,!, & * +"%"$*+,!,,!,'**+('&"& +"$& ,!+"&+"%"$* ,*"& $+*(*'('*,"'&$
+,!($&,'(*'.!'*%
%! %# ]z ]z )F TWO TRIANGLES ARE %"
SIMILAR THEN THE RATIOS OF CORRE SPONDING SIDES ARE EQUAL %! + %" %# + %$ #ROSS 0RODUCTS 0ROPERTY
"+,& &,+ %&,
"++&,+
%&,
*/ & +,!,,!,'**+('&"& +"$& ,!+"&+"%"$*,*"& $+*(*'('*,"'&$
'IVEN ! SECANT SEGMENT CONTAINING THE CENTER OF THE CIRCLE AND A !# zAND ] !$ z TANGENT SEGMENT SHARING AN ENDPOINT OUTSIDE OF A CIRCLE $RAW ]
# # 4HIS IMPLIES A SECANT AND A TANGENT THEREFORE M #!% M !#
!$# IS INSCRIBED THEREFORE M !$# ]z M !# #!% IS FORMED BY
]z
!$# #!% % % THEREFORE N !%# N $%# USING THE !! 3IMILARITY 0OSTULATE 5SING CORRESPONDING SIDES OF SIMILAR TRIANGLES ARE PROPORTIONAL ]z %! ]z %$ #ROSS MULTIPLYING YOU GET %! %# + %$ %#
+!'*%,'(*'.!'*% '*
3TATEMENTS 2EASONS
%$
%!
3EE !DDITIONAL !NSWERS BEGINNING ON P !!
*#&!+" )+#$&!$* "')% #& & +"& *#&!+" #%#$)#+. '*+,$+
&, !*% !+,&&,+%&,
&
!&
'/, ,
, & +, *+-$,+'(*,,'/*!,
(*'('*,!'&!&.'$.!& & '$.'*
+, '*%*'%, !++,!'&,'/*!,&', *
"/% "44&44 3&5&"$)
)-,!'&!&.'$.!&', &
+, *+-$,+'(*,+&,'+'$.'* & '/,'!& #& $+ & *#&!"')%
-"#"#%($#* -"#"#%($#*
%BJMZ)PNFXPSL2VJ[ 4RANSPARENCY !VAILABLE &IND