CAT 2024 Slot 2 · Quant · Question 5

Because CD is parallel to AB, triangles ECD and EAB have equal corresponding angles and are therefore similar.

The ratio of similarity is \frac{CD}{AB}=\frac{1}{2}.

So \frac{ED}{EA}=\frac{EC}{EB}=\frac{1}{2}, which says D is the midpoint of EA and C is the midpoint of EB.

Hence AD=DE and BC=CE.

The trapezium's perimeter is AB+BC+CD+DA=6, and substituting AB=2 and CD=1 gives BC+DA=3.

The perimeter of \triangle AEB is AB+BE+EA, where BE=BC+CE=2\,BC and EA=AD+DE=2\,AD.

So that perimeter is 2+2\left( BC+AD \right)=2+2\times 3=8, which is the third option.

Shortcut. A top-to-bottom ratio of 1:2 makes C and D midpoints, so each slant side of the trapezium is exactly half of the corresponding side of the big triangle. Then BC+DA=6-2-1=3 is the only computation needed, and the perimeter is 2+2\times 3=8.