In the trapezoid ABCD above, BC = 20, and AE = 8. What is the perimete

Solution

Step 1: Analyse Question Stem

• In the given trapezoid ABCD,

o BC = 20
o AE = 8
• We need to find the perimeter of ABCD.

o Now, perimeter of ABCD = AB + BC + CD + DA = AB+ 20+CD + DA

So we need to find AB, CD, and DA.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

Statement 1: AB = 10

• According to this statement: AB = 10

o \(⟹ BE = \sqrt{AB^2 – AE^2} = \sqrt{10^2 – 8^2} = 6\)
• However, with the above information we cannot find AD and CD and hence we cannot find the perimeter.

Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.

Statement 2: AD = 6

• According to this statement: \(AD = 6\)
• However, we don’t know AB and CD. So, we cannot find the perimeter.

Hence, statement 2 is also NOT sufficient and we can eliminate answer Option B.

Step 3: Analyse Statements by combining.

From statement 1:

• AB = 10 and BE = 6

From statement 2:

• AD = 6

On Combining both statements:

• Let’s drop a perpendicular DF, from D to side F, as shown in below diagram.

o So, DF = AE = 8


• From the above diagram, we have, CF = BC – (BE+EF) = 20 – 12 = 8
• Now, in triangle, CDF,

o \(CD = \sqrt{DF^2 + CF^2 }= \sqrt{8^2 + 8^2 }=8\sqrt{2}\)
• Since we know all sides of the trapezoid ABCD, so we can easily find the perimeter of it.

Thus, the correct answer is Option C.

Statement 1: Insufficient
In triangle ABE,
By Pythagoras theorem , BE = 6. so ,EC = 20 - 6 =14
But we still don't know the length of AD and CD , so cant find the perimeter.
Statement 2: Insufficient
We don't know the length of AB and CD.
Can't find the perimeter
On combining,
We can get all sides of trapezium ABCD and its perimeter.


Hence, OA is (C).

From all given information, we can find the length of three sides of the trapezoid. But the rightmost side will still be unknown as we don't know the ratio of angles DCB and ADC to apply the Pythagorean theorem. For perimeter of a trapezoid, length of all 4 sides need to be known.
E is the answer.

In the given figure, it is not mentioned if the Trapezoid is isosceles or not.In that case the only information available with us is that AD||BC,BC=20 and AE=8(height of the trapezium).Draw an imaginary perpendicular from D to BC->DF is perpendicular to BC and DF is 8
1. with this info AB=10, we can find out BE(Pythagorus theorem on triangle ABE, we get BE=6) and EC. but AD and DC cannot be solved for. Not sufficient
2. With this info AD=6, . But with info all we cannot figure out the answer.
Using both the info 1 and 2 we can figure out all the sides as follows
AB=with 1 point , AD=with 2 point, BC=given and DC=(apply pythagorus theorem on rt angel triangle DFC, where DF=8 and FC=BC-(BF)=8

Therefore option C

From 1, in triangle ABE,
By Pythagoras theorem , BE = 6. so ,EC = 20 - 6 =14
But we still don't know the length of AD and CD , so cant find the perimeter.

from 2, we still dont know the length of AB and CD.
So, cant find the perimeter.

From 1 and 2, we know the length of three sides and we can know CD by using the
area of trapezium formula. Lets drop a perpendicular from D to BC and call it as DP.

Area of trapezium ABCD = area triangle ABE + Area triangle DPC + Area rectangle AEPD.
Solve for CD.

Hence , answer is C.

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