Question

Points A(1,0), B(5,0), C(7,6) and D(3,6) are joined to form a quadrilateral ABCD. Point B(5,0) is joined with E(5,6) and A(1,0) is joined with F(1,6). Quadrilateral ABEF is now formed. What is the relation between the perimeter of the quadrilateral ABCD and quadrilateral ABEF?

• A. Perimeter of quadrilateral ABCD < Perimeter of quadrilateral ABEF
• B. Perimeter of quadrilateral ABCD = Perimeter of quadrilateral ABEF
• C. Perimeter of quadrilateral ABCD > Perimeter of quadrilateral ABEF
• D. Perimeter of quadrilateral ABCD =(1/2) Perimeter of quadrilateral ABEF

Answer 1

The correct option is C

Perimeter of quadrilateral ABCD > Perimeter of quadrilateral ABEF

The quadrilateral ABEF is rectangle because BE $\perp$ FC; AF and BE are equal and parallel.
ABCD is a quadrilateral with AB = CD and AB $\parallel$ CD ( A quadrilateral with opposite sides parallel and equal is a parallelogram)
In a triangle, the shortest distance between the vertex and the side opposite to the vertex is the perpendicular distance between them.
Consider, the $\Delta$ AFD, AF$\perp$ FD, AF $<$AD.
Similarly, for the $\Delta$ BEC, BE$\perp$ EC, BE$<$ BC.

Also, AB = EF (since ABEF is a rectangle) and AB = CD (since ABCD is a parallelogram)

Thus, AF + BE$<$ AD + BC
And, AF + BE + AB + EF $<$ AD + BC + AB + EF (Adding AB and EF to both sides)

AF + BE + AB + EF$<$AD + BC + AB + CD (Since AB = EF, and AB = CD, thus EF = CD)

Hence, Perimeter of rectangle $<$ Perimeter of parallelogram

Perimeter of quadrilateral ABEF $<$ Perimeter of quadrilateral ABCD.