We can be drawn the triangle this way for simplicity



Now the question is tricky but if you know some basic rule, it is pretty straightforward.

Two sides are equal, hence this is an isosceles triangle AND it has not necessarily drawn the scale.

The third side \(IK\) must be between 0,1 and 20. Therefore, it can be \(IJ = 1\) and \(JK = 19\) or \(IJ = 19\) and \(JK = 1\) or any combination of them. We do not know.

D is the answer.

Use Mid-Point Theorem: The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.

Now, refer to the figure below;

IH = HG = 5
IJ = JX = x

△IHJ ~ △IGK

i.e. \(\frac{IH}{IG} = \frac{IJ}{IK} = \frac{HJ}{GK}\)

\(\frac{5}{10} = \frac{x}{2x} = \frac{y}{10}\)

\(\frac{1}{2} = \frac{y}{10}\)

\(y = 5\)

Now, in △IHJ
|IH - HJ| < IJ < IH + HJ
|5 - 5| < IJ < 5 + 5
0 < IJ < 10

Hence, option D

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Explanation

First piece of information.

Now here IG = GK. So triangle IGK is an isosceles triangle.

So \(\angle GIK = \angle GKI\).

Second piece of information is H is the midpint of IG anf J is the mid point of IK. From properties of triangle HJ is parallel to GK.


So \(\angle HJI = \angle GKI\).

Further \(GH = HI = 1/2 GI = 5\). Since H is midpoint of GI.

Notice that triangle HIJ is also isosceles triangle with \(\angle HJI = \angle HIJ\) (or \(\angle GIK\)).

So sides HJ = HI = 5.

Hence Quantity A and B are equal. Thus C is the right answer.
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But question asked for IJ?

How the fact that HJ = HI should solve the question, which is about JI?

The question asked about IJ. How will we determine the value of IJ?

How is this possible given that J is the midpoint of IK? I mean IJ and JK must be equal so the case 1 and 19 is not possible. They can be 5 and 5 or 0 and 0 and all the values in between. That's why D is the answer. Am I right?

Since the range of ik is between 0 and 20 if we divide that by 2 we get the range of ij so it is between 0 and 10 so answer d is correct

answer is D
lets take a look we have IJ and GK=10 but what if we predict IK=4 so the and midpoint will be at 2. so ij =2

How is this answer D? Anyone who has done the rudiments know that the answer is C. It clearly states that H and J are midpoints of IG and IK. Is the midpoint of a line not half of the line? And is half of 10 not 5? Why then is IJ being 19? Carcass, please review the answer allocated for the question. Thanks.

The fact that you didnt pick it right , then it does not mean the answer D is wrong.

Please refer to this discussion at gmatclub where the question is the same and the answer is E (D for the GRE exam)

https://gmatclub.com/forum/in-the-figur ... 66564.html

Moreover, refer to this video in which Dabral from Quantum Prep explain in fully details the question and the answer is D

http://quantumgradprep.com/gre-powerpre ... entPage=12 The second video from the top to the bottom.

Regards

The diagram is not drawn to scale.

The side IK = 10 is not given and we cannot assume that it is equal to 10 since j is the midpoint.

The only information as per triangle rule we know the side IK should be less than 20 and greater than zero
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For triangle IGK: IG=GK=10 this means Angles I and K are equal.

Apart from this figure provided details about H as midpoint of IG and J as mid point of IK

So,lets find IK from data given
basic principle for triangle third side: difference between another two sides < third side < sum of two sides
IK can be between 0 and 20

There is no data to exactly determine the length of IJ


option D is correct.

Graphs of any kind (bar, pie, x/y, etc) are drawn to scale on the GRE. Geometric figure are NOT drawn to scale. Therefore, a lot of the time questions like these are really asking "have we given you enough information to clarify the NUMERIC VALUES.

Step by step:

Info 1: H=midpoint, so we now know that IH and HG are equal.

Info 2: J=midpoint, so we now know that IJ=JK. (Note: as of yet, nothing would indicate that these sides are equal).

Info 3: IG=GK=10, so we now know that the left side and the bottom side of the triangle are equal. (Note: this means that we know that the triangle is either an equilateral or isosceles and CANNOT be scalene).

Since there is no info 4, including no further angle measurements, we are left with IK being anywhere from some tiny fraction to a number just below 20 (a number between 0-20, non-inclusive). Since one of the triangle inequalities states that the sum of any 2 sides must be bigger than a third, we know that the max for the side is <(10+10)...or >20. Nothing correlates that side of the triangle to the others. If the prompt told us an angle or gave us general triangular structure information (equilateral vs iso) then we could possible come to a finite value for IK. Without it, we are stuck with a range that is either larger or smaller than 5, and thus, the answer is D.

Personally, I would recommend going through step by step with these type of problems, ESPECIALLY when the answer seems like it should be super easy, as tricking you (like this problem) can work just as well as doing a double fake (making the problem easy...and it looking like there should be a trick but they actually didn't trick you, thus you tricking yourself).

H is the midpoint of IG and J is the midpoint of IK..
We just know that two sides are 10 each but quantity IJ is part of unknown third side..
IK can be anything from 10-10 to 10+10...
\(0<IK<20......\frac{0}{2}<\frac{IK}{2}<\frac{20}{2}.....0<IJ<10\)
So IJ could be less than, equal to or more than 5, depending on IK

Data insufficient to answer

D
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D is correct.

Cant this question solved by the property of similar triangles?
Here, are my 2 cents
Since, IH/IG = IJ/IK = 1/2, IHJ and IGK are similar Triangles.

Please tell me whether I am wrong here

Hi suramya26

I think you misunderstood the values. The answer you got is merely sufficient to prove that the value are divided into two equal parts, which is already stated in the question. Finding the ratio is not sufficient to know whether the value of IJ=5 and IK is 10.

Consider this, if the triangle is right angle triangle, the third side (IK) will be 10 \(\sqrt{2}\), in this case, IJ= 5\(\sqrt{2}\)
Or, If it is an equilateral triangle, the value of IK=10 and IJ is 5

But, as we already have two different values and there could be several other values as well, the value of IJ cannot be determined with the given information.

IMO D

Hope this helps!