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CAT 2022 Slot 1 — QA Question 4

Basics of QuadrilateralsEasy

A trapezium ABCD has side AD parallel to BC. ∠BAD = 90°, BC = 3 cm and AD = 8 cm. If the perimeter of this trapezium is 36 cm, then its area, in sq. cm, is

Answer & solution

Answer: 66

Solution

Easy

Drop a perpendicular from CC to ADAD. Since ∠BAD=90∘\angle BAD=90^\circ and AD∥BCAD\parallel BC, the figure is a right trapezium: ABAB is the height, and the slant side CDCD is the hypotenuse of a right triangle with legs ABAB and (AD−BC)(AD-BC). Use the perimeter to find ABAB, then apply the trapezium area formula.

B A D C E BC = 3 ED = 5 x CD
1

Express the slant side CDCD. Let AB=xAB=x. Foot of perpendicular from CC is EE, so CE=AB=xCE=AB=x and ED=AD−BC=8−3=5ED=AD-BC=8-3=5. By Pythagoras in △CED\triangle CED:

CD=x2+52=x2+25CD=\sqrt{x^2+5^2}=\sqrt{x^2+25}
2

Use the perimeter =36=36 (BC+AD+AB+CDBC+AD+AB+CD) and solve for xx:

3+8+x+x2+25=36x2+25=25−xx2+25=625+x2−50x50x=600 ⇒ x=12\begin{aligned} &3+8+x+\sqrt{x^2+25}=36\\ &\sqrt{x^2+25}=25-x\\ &x^2+25=625+x^2-50x\\ &50x=600\ \Rightarrow\ x=12 \end{aligned}
3

Area of the trapezium =12×(sum of parallel sides)×height=\tfrac12\times(\text{sum of parallel sides})\times\text{height}, with height AB=12AB=12:

Area=12×(3+8)×12=12×11×12=66\text{Area}=\frac12\times(3+8)\times12=\frac12\times11\times12=66
66 sq. cm\mathbf{66}\ \text{sq. cm}
CAT 2022 Slot 1 QA Q4: A trapezium ABCD has side AD parallel to BC. ∠BAD = 90°, BC = 3 cm and AD = 8 cm. If the perimeter of — Solution | TheCATExam