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CAT 2021 Slot 3 — QA Question 12

Basics of QuadrilateralsEasy

Let ABCD be a parallelogram. The lengths of the side AD and the diagonal AC are 10 cm and 20 cm, respectively. If the angle ∠ADC is equal to 30° then the area of the parallelogram is sq. cm. is

Answer & solution

  • A

    25(√3 + √15)/2

  • B

    25(√5 + √15)

  • 25(√3 + √15)

  • D

    25(√5 + √15)/2

Solution

Easy

Drop a perpendicular from AA to side DCDC. In triangle ADEADE the angle at DD is 30∘30^\circ, so it is a 30-60-9030\text{-}60\text{-}90 triangle and we can find DEDE and the height AEAE. Then use the right triangle ACEACE to find ECEC, giving the full base DCDC. The parallelogram area is base ×\times height.

D C A B E AE AC=20 AD=10
1

Resolve the height from the 30∘30^\circ angle. Let EE be the foot of the perpendicular from AA onto DCDC. With ∠ADC=30∘\angle ADC=30^\circ and AD=10AD=10, triangle ADEADE is 30-60-9030\text{-}60\text{-}90.

AE=ADsin⁡30∘=10⋅12=5⇒ DE=ADcos⁡30∘=10⋅32=53\begin{aligned} &AE = AD\sin 30^\circ = 10\cdot\tfrac12 = 5\\ &\Rightarrow\ DE = AD\cos 30^\circ = 10\cdot\tfrac{\sqrt3}{2} = 5\sqrt3 \end{aligned}
2

Use the diagonal in right triangle ACEACE. AE⊥DCAE\perp DC, so ∠AEC=90∘\angle AEC=90^\circ with AC=20AC=20.

EC2=AC2−AE2(Pythagoras)⇒ EC2=400−25=375⇒ EC=375=515\begin{aligned} &EC^2 = AC^2 - AE^2 \quad\text{(Pythagoras)}\\ &\Rightarrow\ EC^2 = 400 - 25 = 375\\ &\Rightarrow\ EC = \sqrt{375} = 5\sqrt{15} \end{aligned}
3

Assemble the base and compute the area. The full base is DC=DE+ECDC = DE + EC, and area == base ×\times height.

DC=DE+EC=53+515(from steps 1, 2)⇒ Area=DC⋅AE=(53+515)⋅5⇒ Area=25(3+15)\begin{aligned} &DC = DE + EC = 5\sqrt3 + 5\sqrt{15} \quad\text{(from steps 1, 2)}\\ &\Rightarrow\ \text{Area} = DC\cdot AE = \bigl(5\sqrt3 + 5\sqrt{15}\bigr)\cdot 5\\ &\Rightarrow\ \text{Area} = 25\bigl(\sqrt3 + \sqrt{15}\bigr) \end{aligned}
Area=25(3+15) sq. cm\text{Area} = 25\left(\sqrt3 + \sqrt{15}\right)\ \text{sq. cm}
CAT 2021 Slot 3 QA Q12: Let ABCD be a parallelogram. The lengths of the side AD and the diagonal AC are 10 cm and 20 cm, respectively. — Solution | TheCATExam