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CAT 2021 Slot 2 — QA Question 21

Basics of QuadrilateralsMedium

If a rhombus has area 12 sq cm and side length 5 cm, then the length, in cm, of its longer diagonal is 

Answer & solution

  • A

    √13 + √12

  • √37 + √13

  • C

    (√13 + √12)/2

  • D

    (√37 + √13)/2

Solution

Medium

The diagonals of a rhombus bisect each other at right angles. Let the half-diagonals be xx and yy. The side gives x2+y2=25x^{2}+y^{2}=25 (Pythagoras) and the area gives 2xy=122xy=12. Add and subtract these to get (x+y)2(x+y)^2 and (x−y)2(x-y)^2; the longer diagonal is 2max⁡(x,y)=(x+y)+(x−y)2\max(x,y)=(x+y)+(x-y).

A B C D O 5
1

Set up the two equations. Let half-diagonals be x=OAx=OA and y=OBy=OB. The right triangle AOBAOB has hypotenuse == side =5=5; the area is 12d1d2=12(2x)(2y)\tfrac12 d_1 d_2=\tfrac12(2x)(2y).

x2+y2=52=25(Pythagoras in △AOB)(1)12(2x)(2y)=12 ⇒ 2xy=12(2)\begin{aligned} &x^{2}+y^{2}=5^{2}=25 \quad\text{(Pythagoras in } \triangle AOB)\quad (1)\\ &\tfrac12(2x)(2y)=12\ \Rightarrow\ 2xy=12 \quad (2) \end{aligned}
2

Combine into perfect squares. Add and subtract (1) and (2).

(x+y)2=x2+y2+2xy=25+12=37 ⇒ x+y=37[(1)+(2)](x−y)2=x2+y2−2xy=25−12=13 ⇒ x−y=13[(1)-(2)]\begin{aligned} &(x+y)^{2}=x^{2}+y^{2}+2xy=25+12=37\ \Rightarrow\ x+y=\sqrt{37}\quad\text{[(1)+(2)]}\\ &(x-y)^{2}=x^{2}+y^{2}-2xy=25-12=13\ \Rightarrow\ x-y=\sqrt{13}\quad\text{[(1)-(2)]} \end{aligned}
3

Solve for the longer diagonal. Adding the two results gives 2x2x, the longer full diagonal (taking x>yx>y).

2x=(x+y)+(x−y)=37+13\begin{aligned} &2x=(x+y)+(x-y)=\sqrt{37}+\sqrt{13} \end{aligned}
Longer diagonal=37+13\text{Longer diagonal}=\sqrt{37}+\sqrt{13}
CAT 2021 Slot 2 QA Q21: If a rhombus has area 12 sq cm and side length 5 cm, then the length, in cm, of its longer diagonal is — Solution | TheCATExam