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Overlapping figures

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Overlapping figures

by mneilans » Thu May 03, 2012 5:16 pm

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HI, the question is

In the figure shon, square CDEF has area of 4. what is the area of triangle ABF

a) 2 root 2
b) 2 root 3
c) 4
d) 3 root 3
e) 6

i have uploaded a picture of the diagram ABF is the triangle on the right....

thanks

Answer is E

Source Gmap prep test 1
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by aneesh.kg » Thu May 03, 2012 6:20 pm
CF = (4)^0.5 = 2

For triangle ABF, the altitude AF = CF(3)^0.5 = 2*(3)^0.5
Since it is a right angle isosceles triangle, AB = base = CF
Area
= (base * height) / 2
= (CF)*(CF)/2
= [(2*(3)^0.5)^2] / 2
= 6 square units

[spoiler](E)[/spoiler] is the correct option
Aneesh Bangia
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by mneilans » Fri May 04, 2012 3:38 pm
Hi there thanks so much for your reply,
are the two triangles similar?

because i managed to get that the left triangles sides would be
2, root 3 (hypoteneuse), and 1 (is this right)
the squares side would be 2

by altitude u mean height right?

why are we multiplying the 2(CF) *root 3 to get the length (what formula is this)

i understand that ABF is an isoceles triangle...and the ratio of sides should be 1:1:root 2 right?

would you mind going into a little more detail
?
thank you so much
marianne
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by aneesh.kg » Fri May 04, 2012 7:20 pm
Hi Marianne,

No, the triangles are not similar.
And yes, 'altitude' and 'height' are the same thing.

In a right angled triangle in which the angles are 90 degrees, 60 degrees and 30 degrees, the ratio of sides is a : b : c = 2 : (3)^0.5 : 1

Image

a : b : c = 2 : (3)^0.5 : 1 is a concise way of writing this:
(i) a : b = 2 : (3)^0.5
(ii) b : c = (3)^0.5 : 1
(iii) c : a = 1 : 2

Note that:

(i) the side opposite to the highest angle (90 deg) is longest and the side opposite to the smallest angle (30 deg) is the shortest.

(ii) This set of ratios is only for a 90-60-30 triangle. Remember them, so that we don't need to go into trignometry.

(iii) These are just ratios. They don't mean that a is 2 units, b is (3)^0.5 units and c is 1 units.

Can you now answer/explain:
(a) how we will apply these ratios in the triangle ABF of your question?
(b) What the area of right angled triangle ABC, with angle B = 90 deg and angle C = 60 deg, and hypotenuse = 6 units?
Aneesh Bangia
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by mneilans » Sat May 05, 2012 3:58 am
Thank you very much
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by Brent@GMATPrepNow » Thu Jan 17, 2019 12:52 pm
mneilans wrote:HI, the question is

In the figure shon, square CDEF has area of 4. what is the area of triangle ABF

a) 2 root 2
b) 2 root 3
c) 4
d) 3 root 3
e) 6

i have uploaded a picture of the diagram ABF is the triangle on the right....

thanks

Answer is E

Source Gmap prep test 1
Image
If the area of the square is 4, then each side has length 2
Image

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 . . .
Image
. . . 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
Image


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

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

What is the area of â–³ABF?
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

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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