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[Smart Math] Geometry Problem 7

Here’s and example of a SMART MATH problem for GEOMETRY.

Problem

The figure ABCD is a parallelogram. Find the perimeter of ABCD.
AB = $2x+2y+7$
BC = $3x+2y-3$
DC = $4x-3y+10$
AD = $x+4y$

1. 22
2. 48
3. 53
4. 59
5. 32

The Usual Method

Since ABCD is a parallelogram, AB = DC and AD = BC.

$2x+2y+7$ = $4x-3y+10$ and $x+4y$ = $3x+2y-3$

$\therefore -2x+5y=3$ and $-2x+2y=-3$

Solving these two equations simultaneously, we get:

x = 3.5 and y = 2

Hence, AB = DC = 2(3.5) + 2(2) + 7 = 18 and,

AD = BC = 3.5 + 4(2) = 11.5

Hence, total perimeter = 2 (18 + 11.5) = 59

(Ans: 4)

Estimated Time to arrive at the answer = 75 seconds.

Using Technique

The perimeter of parallelogram ABCD = sum of length of all sides

= $2x+2y+7$ + $3x+2y-3$ + $4x-3y+10$ + $x+4y$

= $10x+5y+14$

Now, from the options, just subtract 14. This will give us the value of $10x+5y$, which is a multiple of 5 $\because 10x+5y=5(2x+y)$.

Hence, that option which is a multiple of 5 after subtracting 14 from it should be the answer.

22 – 14 = 8      (Not a multiple of 5 and so it is not the correct answer)

48 – 14 = 34    (Not a multiple of 5 and so it is not the correct answer)

53 – 14 = 39    (Not a multiple of 5 and so it is not the correct answer)

59 – 14 = 45    (Is a multiple of 5 and so it is the correct answer)

32 – 14 = 18    (Not a multiple of 5 and so it is not the correct answer)

(Ans: 4)

Estimated Time to arrive at the answer = 15 seconds.
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