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505-555 (Easy)|   Geometry|         
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In the figure above, square CDEF has area 4. 03 Dec 2018, 07:24
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15% (low)

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83% (01:39) correct 17% (02:06) wrong based on 516 sessions

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In the figure above, square CDEF has area 4. What is the area of △ ABF?

(A) 2\(\sqrt{2}\)
(B) 2\(\sqrt{3}\)
(C) 4
(D) 3\(\sqrt{3}\)
(E) 6


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E
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In the figure above, square CDEF has area 4. 03 Dec 2018, 10:48
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IMO E

Triangle \(CFB\) is a 30-60-90 triangle, therefore is side \(FB=2*\sqrt{3}\)...

Triangle \(ABF\) is a 45-45-90 triangle (isosceles right triangle), therefore is side \(BF=AB\) and the hypotenuse is equal to \(2*\sqrt{3}*\sqrt{2}\)

Area of \(ABF=\) \(\frac{1}{2}*2*\sqrt{3}*2*\sqrt{3}=\)\(6\)
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In the figure above, square CDEF has area 4. 17 Jan 2019, 12:33
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HKD1710


In the figure above, square CDEF has area 4. What is the area of △ ABF?

(A) 2\(\sqrt{2}\)
(B) 2\(\sqrt{3}\)
(C) 4
(D) 3\(\sqrt{3}\)
(E) 6


Show Spoiler
Attachment:
ABF.JPG
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If the area of the square is 4, then each side has length 2


At this point, we have a special 30-60-90 right triangle. When we compare this blue triangle to the BASE 30-60-90 right triangle . . .

. . . we see that the blue triangle TWICE the size of the BASE 30-60-90 right triangle

So, These are the measurements of the blue triangle



Finally, we have special 45-45-90 right triangle.


This triangle is also an ISOSCELES triangle, so the other side ALSO has length 2√3


What is the area of △ABF?
area of a triangle = (base)(height)/2

So, area of △ABF = (2√3)(2√3)/2
= (4√9)/2
= (4)(3)/2
= 12/2
= 6

Answer: E

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Brent­
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In the figure above, square CDEF has area 4. 10 Feb 2019, 21:20
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HKD1710


In the figure above, square CDEF has area 4. What is the area of △ ABF?

(A) 2\(\sqrt{2}\)
(B) 2\(\sqrt{3}\)
(C) 4
(D) 3\(\sqrt{3}\)
(E) 6

Show more

First let us calculate the side CF through the given area of square
side becomes 2

After this triangle CFB will be a 30 60 90 triangle

From here we can get the corresponding side FB as 2 \(\sqrt{3}\)

Now since △ ABF, is an isosceles triangle, the height and base will be equal

1/2 * 2 \(\sqrt{3}\) * 2 \(\sqrt{3}\)

6

E
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In the figure above, square CDEF has area 4. 06 Feb 2023, 15:40
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