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CAT 2018 Slot 2 — QA Question 17

Basics of QuadrilateralsEasy

A parallelogram ABCD has area 48 sqcm. If the length of CD is 8 cm and that of AD is s cm, then which one of the following is necessarily true?

Answer & solution

Correct answer: s ≥ 6

  • A

    s ≠ 6

  • s ≥ 6

  • C

    5 ≤ s ≤ 7

  • D

    s ≤ 6

Solution

Easy

The area fixes the height dropped onto side CDCD. Side ADAD is the slant side, which is the hypotenuse of a right triangle with that height as one leg — so ADAD can never be shorter than the height.

1

Find the height. Area =base×height=\text{base}\times\text{height} with base CD=8CD=8.

48=8⋅h⇒ h=6\begin{aligned} &48=8\cdot h\\ &\Rightarrow\ h=6 \end{aligned}
2

Relate ADAD to the height. Drop a perpendicular of length h=6h=6 from AA onto line CDCD. Then AD=sAD=s is the hypotenuse of a right triangle with legs hh and the horizontal offset d≥0d\ge 0.

s=h2+d2=36+d2\begin{aligned} &s=\sqrt{h^2+d^2}=\sqrt{36+d^2} \end{aligned}
3

Bound ss. Since d2≥0d^2\ge 0, with equality only when the parallelogram is a rectangle (s=6s=6).

s=36+d2≥36=6⇒ s≥6\begin{aligned} &s=\sqrt{36+d^2}\ge\sqrt{36}=6\\ &\Rightarrow\ s\ge 6 \end{aligned}
s≥6s\ge 6

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CAT 2018 Slot 2 QA Q17: A parallelogram ABCD has area 48 sqcm. If the length of CD is 8 cm and that of AD is s cm, then which one of t — Solution | TheCATExam