In the floor plan of an executive's beach house above, the north and

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Solution

Given:
Let us name the point in the diagram as shown.

• AE = 30 ft
• BE = 30 ft
• CF = 15 ft
• AD = 30 ft
• CD = 30 ft

To Find:

• Area of Bedroom or ∆ ACD

Approach and Working:

Since EF is parallel to BC, So, two corresponding angles of ∆ AEF and ∆ ABC will be equal.

• ∆ AEF and ∆ ABC are similar triangles.
• Therefore, \(\frac{AE}{AB}\) = \(\frac{AF}{AC}\)
• \(\frac{30}{30 +30}\)= \(\frac{x}{x+15}\)

o x = 15
o AC = x+ 15 = 30 ft

• Hence, ∆ ACD is an equilateral triangle since all the sides are equal.

o Thus, Area of ∆ACD = \(\frac{√3}{4} * 30^2\) = 225√3

Therefore, option C is the correct answer.

Correct Answer: Option C
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Make an parallel to the titled side of the living room from 30 feet (Middle of 60 feet)

Using similarity, the third side of the triangle=2*15=30ft.

Area of the triangle=\(\sqrt{3}/4*30*30=225\sqrt{3}\) IMO, option C.

We see that a side of the smaller triangular kitchen is exactly half of the larger triangle that comprises of the bath, kitchen and living room (30 ft vs. 60 ft). Since the two triangles are similar, then the side of the kitchen that opens to the bedroom is 15 ft since the wall that borders the living room and the bedroom is 15 ft. Thus, the bedroom is in the shape of an equilateral triangle with side length of 30 ft. Therefore, the bedroom has an area of (30^2 x √3)/4 = 900√3/4 = 225√3 square feet.

Answer: C
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Hi Scot/Payal,

What are we not subtracting the area of the "shark fin" portion? Response would be appreciated.

Mid point theorem, a line joining the mid point of two sides of a triangle is parallel and 1/2 the third side. Converse is also true.

Quora: https://www.quora.com/What-is-the-proof ... nt-theorem

So, the bedroom is basically an equilateral triangle with sides 30. Now we can easily find the area.

EgmatQuantExpert Bunuel VeritasKarishma

I think the answer should be E unless the Q clearly specifies that the "shark fin" is part of the bedroom. We wouldn't have cared about the area of "shark fin" if we had 2 "shark fins" connected to the bedroom, one outside and one inside so that they could have cancelled each other out.

From the tags, it seems to be an official Q but I'm surprised that an official Q has this kind of ambiguity.

Any thoughts on that? Thanks!

The “shark fin” portions in the diagram tell us that there’s a door there and also tells us in which direction the door is opening. It’s the way technical drawings for floor plans are made. Considering also that we are told neither about the dimensions of the “shark fin” parts nor that those sections are actually quarter circles as they look, we can safely ignore those portions.
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We can use mid point theorem. The side of the kitchen that opens to the bedroom is 15 ft, since the wall that borders the living room and the bedroom is 15 ft. Thus, the bedroom is in the shape of an equilateral triangle with side length of 30 ft. Therefore, the bedroom has an area of (30^2 x √3)/4 = 900√3/4 = 225√3 square feet.

This is the most screwed-up floor plan one could imagine! Why on Earth would you put a door between the kitchen and the bedroom??! Plus good luck with the bed in that triangular bedroom!

@experts could you explain how we used the midpoint theorem here? :$ how do we know that the north wall is cutting the bedroom side wall at the mid point ?

Kritisood
Please refer ShukhratJon's Solution for your answer.(According to the mid-point theorem, the line joining the mid-points of two sides of a triangle is parallel to the third side of the triangle. However, note that we are given that line DE is || to AC and lengths of AD = DB = 30. So, we have to apply the concept in a reverse manner.)

But i think you can solve the question without applying the similar triangles/midpoint theorem/concept.
Consider the rough diagram below. Consider the triangle △BCF. CE = 15, BF = CF = 30


Now if ∠ECF = 60° and on joining EF where EF⟂BC, we have right △FEC with ∠EFC = 30°. However, this has to be proved.(Note that this is about applying reverse concept and its a little time consuming.)

In a 30-60-90 right triangle the ratio of sides is \(x:\sqrt{3}x:2x\).
Here, if △BCF is a 30-60-90 right triangle, then sides ratio is CE:EF:CF = 15:EF:30.
Thus, \(EF = 15\sqrt{3}\)
Therefore, with this logic you can prove BE(△BEF) = 15, implying △BCF is an equilateral triangle with side length of 30.

From here onwards the area of an equilateral triangle = \(\frac{\sqrt{3}}{4}*x^2\) where x is the side of the triangle.
Area = \(\frac{\sqrt{3}}{4}*30^2\)
= \(225\sqrt{3}\)

Answer C.
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Answer: Option C

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Mid segment Theorem:

A line drawn from the mid-point of one side of a triangle to the mid-point of the opposite side of the triangle will be:

(1)parallel to the third, parallel side

And

(2)Half the length of that third, parallel side

This means that the Unknown Distance to complete the 3rd side of the Bedroom is 15

Equilateral triangle with side length 30

Area = (30)^2 * sqrt(3) * (1/4) =

225 * sqrt(3)

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How we actually know that a combination of Bathroom Kitchen and Living room will be a triangle. The question doesn't tell us that a side of Bathroom and Kitchen is on the same line.
How can we assume that a combination of Bathroom Kitchen and Living room is a triangle

Video solution from Quant Reasoning:
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But what if the south wall of the bath is not in the same horizontal line as the south wall of the living room?

If a line looks like a line, it’s a line. panpan19

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