pramodmundhra15 wrote:
Can't we use the formula of a trapezoid and solve this question ?
which is 1/2*sum of parallel sides*height.
Hello
pramodmundhra15Yes, we absolutely can use the formula of trapezoid to solve the problem. But we will have to use an inference by drawing up a triangle. Let me illustrate to you what I mean.
SOLVING THE PROBLEM USING TRAPEZOID FORMULAThe figure in the question is divided into 4 identical trapezoids with equal area.
- The total area of the rectangle = Length * Width
- = (20 * 12) * 6 = 240 * 6 inch2 [Converting 20 feet into inches by multiplying by 12]
- So, sum of areas of all 4 trapezoids = 240 * 6 inch2 [Since area of rectangle = sum of areas of all 4 trapezoids
- Hence, area of each trapezoid = (240 * 6)/4 = 360 [Since each trapezoid had the same area] -----(1)
Now, let’s try to get the area of one trapezoid using a different way – using the formula for area of trapezoid. Consider the trapezoid with one side AB: (We have named the side parallel to AB as DE.)
- Area of trapezoid ABED = ½ * (Sum of parallel sides) * (Height between parallel sides)
- = ½ * (AB + DE) * BE
- = ½ * (AB + DE) * 6 [Given that width = 6 inches and BE = width)
- = 3 * (AB + DE) ------------- (2)
Combining (1) and (2), we get: 3 * (AB + DE) = 360, or
(AB + DE) = 120 ----- (3)
Now, recall that the question wants us to find AB, but we cannot get that from equation (3) alone, since there are 2 unknowns AB and DE. Hence, we must find another equation relating AB and DE. See that we haven’t yet used the 45 degree angles given – it’s time!
Let’s again enter trapezoid ABED. Drop a perpendicular from D on line AB to create triangle ADF .
- In triangle AFD:
- Angle A = x = 45 degrees and angle AFD = 90 degrees. Hence, angle FDA is also 45 degrees, making triangle AFD a 45-45-90 triangle.
- Thus, AF = FD = 6. [Sides opposite equal angles are equal, and FD = BE = 6] --- (4)
- Finally, observe the above figure to infer that AB = AF + FB and FB = DE.
- Thus, AB = AF + DE
- AB = DE + 6 [from (4)]
- Or DE = AB – 6 ------ (5)
From (3) and (5), (AB + AB – 6) = 120
- (2AB – 6) = 120
- 2AB = 120 + 6
- Therefore, AB = 126/2 = 63 inch
Converting it into feet we get, AB = 63/12 =
5 feet 3 inches.So, you see that we could use the formula of trapezoid, but to draw the relationship between AB and DE, it was helpful to draw the triangle.
Hope it helps!
Best Regards,
Ashish Arora
Quant Expert,
e-GMAT