Question

What is the length of AC?

Answer 1

Since both triangles have a right-angle, and the two angles at vertex C are also twins, thus both triangles are similar by AA, therefore,

$\bf \cfrac{small}{large}\qquad \cfrac{DE}{BA}=\cfrac{EC}{CA}\implies \cfrac{7}{84}=\cfrac{x}{156-x}\implies 1092-7x=84x \\\\\\ 1092=91x\implies \cfrac{1092}{91}=x\implies 12=x\qquad \quad \qquad \boxed{AC=156-12}$