The trapezoid shown in the figure above represents a cross

The trapezoid shown in the figure above represents a cross section of the rudder of a ship. If the distance from A to B is 13 feet, what is the area of the cross section of the rudder in square feet?

(A) 39
(B) 40
(C) 42
(D) 45
(E) 46.5

The formula for calculating the area of a trapezoid is \(Area=\frac{1}{2}(base_1+base_2)(height)=\frac{1}{2}(2+5)(height)\).

So, we need to find the height AC: \(AC=\sqrt{AB^2-BC^2}=\sqrt{13^2-5^2}=12\).

Therefore, \(Area=\frac{1}{2}(2+5)*12=42\).

Answer: C.
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See Image for half triangle + rectangle solution.

Rotate the figure:Hope it's clear.
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You could save yourself some calculation here if you knew the pythagorean triple 5-12-13. That would shave a few seconds off of the problem.

Although this problem may seem confusing based on the description of the shape, we must keep in mind that the diagram provided is simply a trapezoid. So when we are asked to determine the area of the “cross section of the rudder,” this really means "What is the area of the trapezoid?"

The formula for the area of a trapezoid is:

area = (base 1 + base 2) x height/2

The two bases are given as 2 feet and 5 feet. Because the height is always perpendicular to the base, the height of this particular trapezoid is the side that begins at A and ends at the right angle located at the bottom right of the figure. We don't know this value, so we must calculate it. We could use the Pythagorean Theorem to determine the height, but instead we note that we have a 5-12-13 right triangle, where the unknown side (or the height) is 12.

We can now substitute the values for the two bases (2 and 5) and the height (12) into our area equation.

area = (base 1 + base 2) x height/2

area = (2 + 5) x 12/2

area = 7 x 6

area = 42

Answer is C.
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Let's add a few things to the diagram to make the solution easier to follow.


Okay, first we should recognize that the two 90-degree angles (at vertices A and D) mean that sides AC and BD are PARALLEL, which means the rudder (quadrilateral ACBD) is a TRAPEZOID.

Area of trapezoid = (sum of the parallel sides)(distance between parallel sides)/2
So, all we need to do now is determine the length of side AD.

Now recognize that ∆ABD is a RIGHT TRIANGLE.
So, we can use the Pythagorean Theorem to get the equation: 5² + (AD)² = 13²
Evaluate: 25 + (AD)² = 169
Simplify: (AD)² = 144
Solve: AD = 12

Aside: We could have saved time be recognizing that this is a 5-12-13 (since we already knew the 5 and 13 lengths)

Okay, now that we know the length of side AD, we can find the area of this trapezoid.

Area = (2 + 5)(12)/2
= 42

Answer: C

Cheers,
Brent

Well I don't know the formula for calculating the area of trapezoid. Also, I like to solve sums without using paper pen and with few basic formulae.

I first found out the base (the bottomline in the question figure) using pythogoras triplet (5-12-13).
Then I imagined the figure as right-angled triangle sitting on top of a rectangle.
A(Triangle) = 1/2*3*12 +
A(Rectangle) = 2*12
Total = 42.
I don't know how to insert figures here so can't explain properly

Here's the easiest way.

Because we are given so much info, we can treat the one shape as two distinct shapes:

First - a rectangle with dimension of 12*2

Second - a right triangle with legs 3 and 12

Solve for the distinct areas of either to produce 18 and 24 respectively.

Add to give 42

why AC is height here??? couldnt get it

If the trapezoid represents a cross section of the rudder of a ship, then the area of the cross section of the rudder in square feet should have been the area of both the triangles represented in the figure. I know I am obviously wrong here but the way I mentioned is stuck in my head.

That's not wrong at all. But the area of a trapezoid can also be found with the direct formula as shown in posts above. If you expand that formula you'll see that you basically get the sum of the areas of the two triangles.

Does this make sense?
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Oh right. Thank you. That is some relief definitely !

Wow ..Neat design

So what I am taking from this is that as long as we know it's a trapezoid, then we can use that area formula. In other words, it must be stated in the question. Visually, that looks nothing like a trapezoid (maybe half of one).
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