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CAT 2018 Slot 1 — QA Question 19

Basics of QuadrilateralsEasy

Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, in cm, the possible length and breadth of ABCD?

Answer & solution

Correct answer: 24, 10

  • A

    24, 12

  • 24, 10

  • C

    25, 10

  • D

    25, 9

Solution

Easy

A rectangle inscribed in a circle has its diagonal equal to the diameter. So the length and breadth must satisfy l2+b2=(diameter)2l^2+b^2 = (\text{diameter})^2. Test the options against this.

1

Diagonal == diameter. The diagonal of the rectangle is the diameter of the circle:

diagonal=2×13=26⇒ l2+b2=262=676(Pythagoras on the diagonal)\begin{aligned} &\text{diagonal} = 2\times 13 = 26\\ &\Rightarrow\ l^2 + b^2 = 26^2 = 676 \quad\text{(Pythagoras on the diagonal)} \end{aligned}
2

Test the candidate pairs.

242+122=576+144=720≠676242+102=576+100=676✓252+102=625+100=725≠676252+92=625+81=706≠676\begin{aligned} &24^2+12^2 = 576+144 = 720 \neq 676\\ &24^2+10^2 = 576+100 = 676 \quad\checkmark\\ &25^2+10^2 = 625+100 = 725 \neq 676\\ &25^2+9^2 = 625+81 = 706 \neq 676 \end{aligned}
Length, breadth=24, 10 cm\text{Length, breadth} = \mathbf{24,\ 10}\ \text{cm}

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CAT 2018 Slot 1 QA Q19: Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, — Solution | TheCATExam