GMAT: relative areas of triangles?

Question

If I have two equally proportioned triangles (same angles, ratio of sides etc), if the area of one triangle is TWICE that of the other, what is the length of the base (S) of the large one in terms of the smaller (s).

I'm presuming triangle shape is immaterial; it's just that they're equivalent and one is 2x the area of the other; and presumably the answering equation relates to any of the sides you'd wish to compare.

The answer is S = (sqrt of 2) x s.

But how do I know this? Pls advise how I could have worked it out. I have no further data (ie about height etc).

Thanks.

Answer 1

Ok... Assume these to be true...

A = (S * H) / 2
a = (s * h) / 2

Since A has an area that is two times bigger than a, this must also be true...

A = 2 * a

So...

(S * H) / 2 = s * h

However, remember you said that the ratio of the sides was the same... H and h are a certain number... "n" in this case bigger than S and s... So...
H = n * S
h = n * s

Substitute:

S * (n * S)/2 = s * (n * s)

You end up with:

S^2 * n / 2 = s ^ 2 * n

Multiply by 2 and divide both sides by "n" (n can't be zero anyways).

S ^ 2 = 2 ( s ^2 )

Take the square root of both sides...

S = sqrt(2) * s

Hope this helps. :)

Answer 2

Sonny, if this is not a homework question here is a far easier answer.
Whatever is true for a triangle is true for two triangles joined.
Some triangles join on the hypotenuse to make a square.
We can change the problem therefore to two squares, one half the area of the other.
A=S^2 and a=s^2
if A = 2a then S^2=2s^2
taking square root of both sides
S = Root 2 x s

Answer 3

Hi,

When you have two equally proportioned triangles (same angles, ratio of sides etc), if the ratio of the length of a side in the larger figure to the the length of the corresponding side in the smaller figure is a:b, then the ratio of all corresponding lengths will also be a:b. Their perimeters would have the ratio a:b. Corresponding diagonals would have the ratio of a:b for their lengths.

That changes when you go to area. Think of a rectangle for an example. If their bases had a ratio of a:b and their heights had a ratio of a:b, then when you multiply base * height it would be like multiplying their ratios together (a:b)*(a:b) which means the ratio of their areas would be a²:b².

You can see this is true with an example. Suppose 2 similar right triangles have sides of 3,4,and 5 in the smaller triangle and 6,8, and 10 in the bigger triangle. Clearly the ratio of their corresponding sides is 3/6 = 4/8 = 5/10 = ½. 1:2 is the ratio of the smaller side to the larger side. If you find the area of each triangle by A = ½bh, you get A = ½(3)(4) = 6 and A = ½(6)(8) = 24. The ratio of their areas is 6:24 or 1:4. This is a²:b².

So when you have the ratio of their AREAS is 2:1, this is a²:b².

If a²:b² = 2:1,
..................__...__
then a:b = √(2):√(1) which simplifies to
.........._
a:b = √2:1

I hope that helps!! :-)

Answer 4

Area depends on square of dimensions.

Area (2 times), dimension (square root of 2) times

Answer 5

these are similar triangles therefore we can do this
abig/asmall=(lengthbig/lengthsmall)^2
2/1=(lengthbig/lengthsmall)^2
therefore the ratio is sqrt2