In the given diagram, We are given two right triangles BAC and DEC.
Triangle BAC has right angle at A and triangle DEC has right angle at E.
Also we are given <BCA ≅<DCA.
Therefore,
Triangle BAC is similar to triangle DEC by Angle Angle similarity theorem.
Note: The sides of similar triangles are in proportion.
Therefore,
[tex]\frac{AB}{ED} = \frac{AC}{EC}[/tex]
[tex]\frac{84}{7}=\frac{156-x}{x}[/tex]
[tex]\mathrm{Apply\:fraction\:cross\:multiply:\:if\:}\frac{a}{b}=\frac{c}{d}\mathrm{\:then\:}a\cdot \:d=b\cdot \:c[/tex]
[tex]84x=7\left(156-x\right)[/tex]
[tex]84x=1092-7x[/tex]
[tex]\mathrm{Add\:}7x\mathrm{\:to\:both\:sides}[/tex]
[tex]84x+7x=1092-7x+7x[/tex]
[tex]91x=1092[/tex]
[tex]\mathrm{Divide\:both\:sides\:by\:}91[/tex]
[tex]\frac{91x}{91}=\frac{1092}{91}[/tex]
[tex]x=12[/tex]
AC = 156 -x.
Plugging value of x in 156-x, we get
156-12 = 144.
Therefore, AC = 144 units.
Correct option is D. 144.
