You are saying yourself that you assumed that it must be 30-60-90 triangle. Why? Do you have any reasons supporting this claim? No.

So, you assumed with no ground for it that the lengths of the sides are integers. Knowing that hypotenuse equals to 10 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple - 6:8:10. Or in other words: if \(a^2+b^2=10^2\) DOES NOT mean that \(a=6\) and \(b=8\), certainly this is one of the possibilities but definitely not the only one. In fact \(a^2+b^2=10^2\) has infinitely many solutions for \(a\) and \(b\) and only one of them is \(a=6\) and \(b=10\).

Hope it's clear.
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1 is clearly insufficient.
2. the ratio is given as 1:2 again insufficient.
combining both
using Pythagoras theorem.
10^2=x^2+(2x)^2
100= 5x^2
this should give us x followed by the length of the side of the larger square.
Therefore C

C

1)side of small sq is given but no info about large sq. Insuff
2)side of small sq not given and info about large sq is there. insuff

suppose side of large sq is 3x so its divided into x and 2x and side of small square is given(which acts as hypotenuse of all4 triangles)

apply pythagos. theorem and get value of x

There is one unknown, that is the side of the larger square.
We need something that gives us the side of the bigger square that is in either a variable form which could be determined or in a direct value form.
Statement 1 does not have any relation to side of bigger square, hence insufficient.
Statement 2 gives a variable relation as, if side of bigger square is 3x, then the triangles whose hypotenuse is given by Statement 1 has sides 2x and x which gives an equation to determine x (4x^2 + X^2 = 10^2).
Therefore both statement are required together to arrive at a definite answer.

For statement (1) why can't we assume this is a common triangle?

We know the hypotenuse is 10 and that it is a right triangle. With this information the side lengths of the large triangle should be 6 & 8.

Could someone please elaborate on this a bit? I also put choice one, because I assumed that the triangles formed in the corners were 30,60,90 triangles. Therefore, I ended up picking A because I had just assumed that you can back into x,x\sqrt{3},2x where 2x=10.

Clear as day, thank you. I guess sometimes I force myself to try to see things that aren't there with DS. Thanks Bunuel

@Bunnel,

Why can't we have answer as 'A'
My Intake:

st1: Side length of smaller square is 10 cm.
Can't we say area of smaller square=2*area of larger square
10*10=2 S^2
S^2=50
S=root 50

Is this possible?
Please help!!!

How?
If it is mentioned in the question then only you can consider it, by default you are not allowed to assume anything.

Guys one question. if we know that it is not a right angle , why did u used phythagorean theorem ?

I'm a bit confused.

Bunuel
Are we to assume that the ratio of each side is in line with the dimensions formed by the verticie?


This is the assumed construction

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But this is the construction if we randomly split each side.

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1. We have one number - 10 as the hypotenuse and we know that its a square. Nothing can be done with this info

2. Now this statement gives the ratio of the sides. We have a ratio with respect to the larger sqaure and no other info on the larger square.

combining, we'd get the hypotenuse and the side ratios, hence sufficient to figure out the side of the larger square

I liked the question you asked. Because I had read somewhere that the DS questions do not have the figure drawn to scale. It got me thinking about the possibilities. Then I realized that since the figure inside is a square, we should get consistent lengths of each of the sides which will not be the case if we interpret the statements in the way shown in your figure. So the only possibility is one side being x and the other one being 2x.

@I have a quick question. In this problem, it's said that each of the vertices of the smaller square cut one side of the larger square into the ratio 1:2. In that case, it might even be that any right triangle's equation be, 'x^2 + x^2 = 10' or '2x^2 + 2x^2 = 10' since it is not given which part of the line segment of the larger square gets x and which gets 2x. In that case, my answer changes again. Then, how can I take C for an answer? Since it will not be a definite answer, my answer must be E? Bunuel

Here is a tip. Try constructing the case you are describing.
.

Assign angles and see what you get.
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Hi Bunuel,

I have the same query. If I calculate x based on the figure you shared, I get 3 values of x i.e.
1) x = \(5 \sqrt{2}\)
2) x =\( \sqrt{10}\)
3) x = \(\frac{5}{\sqrt{2}}\)

Since there are 3 different values of x, does it not imply that we do not have a unique solution, hence, E?
Or, do we take the figure in the question at its face value (plus, the 1:2 condition in B) and be assured that it is. a 30-60-90 triangle?

Will really appreciate it if you could let me know where I am going wrong.

ANswer: Option C

Please check the video for the step-by-step solution.

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(1) Hypotenuse of the right triangle is 10; two other sides can take several values; Insufficient.

(2) Two sides of the large square are: \(1x:2x;\) x can take numerous values; Insufficient.


Using Both information:
\(x^2+(2x)^2=10^2\); Gives specific value; Sufficient.

The answer is C
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