Answer:
See working below
Step-by-step explanation:
You can apply Thales Intercept Theorem ("similar triangles")
a. [tex]\frac{36}{24}\:=\:\frac{48}{ET},\:ET\:=\:\frac{24\cdot 48}{36}\:=\:32[/tex]
b. [tex]\frac{36+24\:}{AJ}=\:\frac{48+ET}{40},\:\frac{60\:}{AJ}=\:\frac{48\:+\:32}{40},\:\frac{60}{AJ\:}=\:\frac{80}{40},\:AJ\:=\:\frac{1}{2}\left(60\right)\:=\:30[/tex]
c.
[tex]\\\frac{48}{CT}\:=\:\frac{36}{36+24}\:=\:\frac{27}{AT},\\\\CT = CE + ET = 48 + 32 = 80\\\\\frac{48}{80}\:=\:\frac{36}{60}\:=\:\frac{27}{AT},\:0.6\:=\:\frac{27}{AT},\:AT\:=\:\frac{27}{0.6}\:=\:45[/tex]
d.
[tex]\frac{36+24+AJ}{12\:}=\:\frac{48+ET+40}{OS}\\\\\frac{36+24+30}{12}=\:\frac{48+32+40}{OS},\:\frac{90}{12}\:=\:\frac{120}{OS},\:OS\:=\:\frac{12\cdot 120\:}{90}=\:16[/tex]
e.
[tex]\frac{36+24}{36+24+AJ\:}=\:\frac{48+ET}{48+ET+40\:}=\:\frac{AT}{JO}\\\\\frac{60\:}{36+24+30}=\frac{\:48+32}{48+32+40}\:=\:\frac{45}{JO}\\\\\frac{60}{90}=\:\frac{80}{120}\:=\:\frac{45}{JO}\\\\JO\:=\:\frac{135}{2}\:=\:67.5[/tex]
f.
[tex]\frac{36+24+AJ\:}{36+24+AJ+12}=\frac{\:48+ET+40}{48+ET+40+OS}\:=\:\frac{JO}{NS}\\\\\frac{90}{102}\:=\:\frac{120}{136}\:=\:\frac{67.5}{NS}\\\\NS\:=\:\frac{136\cdot 67.5}{120}=\:76.5[/tex]
