When can you assume information in geometry problems?

[NathanielJ.Ho]

If CD is the diameter of the circle, does x equal 30?

(1) The length of CD is twice the length of BD.

(2) y = 60



Solution:
Triangle DBC is inscribed in a semicircle (that is, the hypotenuse CD is a diameter of the circle). Therefore, angle DBC must be a right angle and triangle DBC must be a right triangle.

(1) SUFFICIENT: If the length of CD is twice that of BD, then the ratio of the length of BD to the length of the hypotenuse CD is 1 : 2. Knowing that the side ratios of a 30-60-90 triangle are 1 : : 2, where 1 represents the short leg, represents the long leg, and 2 represents the hypotenuse, we can conclude that triangle DBC is a 30-60-90 triangle. Since side BD is the short leg, angle x, the angle opposite the short leg, must be the smallest angle (30 degrees).

(2) SUFFICIENT: If triangle DBC is inscribed in a semicircle, it must be a right triangle. So, angle DBC is 90 degrees. If y = 60, x = 180 - 90 - 60 = 30.

The correct answer is D.

I answered A because I did not realize that I could assume the triangle was inscribed in the circle. On my last practice exam I assumed the triangle was a right triangle and answered incorrectly. Here is that question

What is the radius of the circle pictured to the right?

(1) The measure of arc PQ is 4.

(2) The center of the circle is at point O.


Solution:



We are given a diagram of a circle with point O in the interior and points P and Q on the circle, but are not given any additional information. We are asked to find the value of the radius.

(1) INSUFFICIENT: This statement tells us the length of arc PQ but we are not told what portion of the overall circumference this represents. Although angle POQ looks like it is 90 degrees, we are not given this information and we cannot assume anything on data sufficiency; the angle could just as easily be 89 degrees. (And, in fact, we're not even told that O is in the center of the circle; if it is not, then we cannot use the degree measure to calculate anything.)

(2) INSUFFICIENT: Although we now know that O is the center of the circle, we have no information about any actual values for the circle.

(1) AND (2) INSUFFICIENT: Statement 2 corrected one of the problems we discovered while examining statement 1: we know that O is the center of the circle. However, we still do not know the measure of angle POQ. Without it, we cannot determine what portion of the overall circumference is represented by arc PQ.

The correct answer is E.

Here I answered A because I assumed it was a right angle and the circumference was 16pi. When can you assume and when can you not in geometry problems? Is there a general rule?

[Kweku.Amoako]

Hi,

I am not sure you can ever assume any information that is not specifically given in words or in a diagram. Eg I can not assume that just because a shape looks like a square automatically means it is a square. You to be able to justify myunderstanding of any diagram. Eg in the case of the square. I will need information such us " all angles are 90 and all sides are equal in length" or " the question should state that it is square" or this information could be shown on the diagram itself.

Infact if you had not made that assumption on the first daigram, you would have probably guessed the right answer. The angle looks like a 90 degree angle but no where in the question stem or on the diagram can you justify this assumption. The main trap for that question was the assumption you made. We could set up an equation for the first diagram

(1) r = [4/(2*pi)]/(360/x) so to find r we need will need x
(2) Not much info

both combined, we still don't know what x is hence E.

Likewise in the second diagram the stem of the tells us cd = diameter, which also means angle CBD = 90, there x+y = 90 and x = 90 - y. This means to find x we need the value of y.

1) CD = 2BD. since we already know this a right angle triangle, i know this is a 1:root(3): 2 or 30:60:90 triangle. So I know x = 30 . SUFFICIENT

2) Y = 60 since we know the value y, we kow x = 90 - 60 = 30. Sufficient

So to answer your question, no do not make any assumptions unless you can justify from the information given

[Ben Ku]

Hi Nathaniel,

You can only assume information that is explicitly stated in the question. The figure just provides an idea for us to understand how the pieces are related to each other.

In your first example, the triangle is inscribed in the circle, meaning all three vertices of the triangle are on the circle itself. In this case, the question tells us that CD is the diameter (so therefore we can conclude that it's a right triangle). However, if the question had NOT given us the information that CD is the diameter, then we should NOT conclude it's a right triangle. (The rule is: if one of the legs of an inscribed triangle lies on the diameter of the circle, then the triangle is right. Not every inscribed triangle is right).

In your second example, the picture looks like it's a POQ is a right angle. In fact, point O looks like the center of the circle. However, the question does not provide either of those two pieces of information. All we know from the picture is that point O is somewhere inside the circle, and P and Q lie on the circle.

Hope that helps.

[hummcon]

Hi Nathaniel,

You can only assume information that is explicitly stated in the question.
In your first example, the triangle is inscribed in the circle, meaning all three vertices of the triangle are on the circle itself. In this case, the question tells us that CD is the diameter (so therefore we can conclude that it's a right triangle). However, if the question had NOT given us the information that CD is the diameter, then we should NOT conclude it's a right triangle. (The rule is: if one of the legs of an inscribed triangle lies on the diameter of the circle, then the triangle is right. Not every inscribed triangle is right).

I don't think that you can assume that this triangle is inscribed. Because it doesn't say that anywhere in the problem. The points may or may not be located on the circle. Maybe it's a tad-bit too loosely written? However, I'm open to being challenged.

Here is an example that is much more clear about what the parameters are - also from the MGMAT CAT tests. I had no problem answering the following problem correctly, but the above problem really threw me for a curve for a while, until I just shrugged it off and realized the question could have been written more clearly.

For the triangle shown above, where A, B and C are all points on a circle, and line segment AB has length 18, what is the area of triangle ABC?

(1) Angle ABC measures 30°.

(2) The circumference of the circle is 18.

[jlucero]

The original problem does say that CD is the diameter of the circle, so we at least know that those two points must be on the circle. However, I will grant that knowing B is on the circle is essential to making each statement sufficient. Official GMAT problems will always be explicit about these types of things. That said, diagrams are generally there to give you an idea of what the figures should look like but not there to try to trick you. You can generally use them to visualize what a figure will look like (BCD all look like points on the circle), but you should never use them to deduce angles UNLESS they give you a piece of information to use (i.e. CD is the diameter).

As a side note, which test is the question you added from? And what is the problem name? It is impossible that a line inside of a circle could be the same length as the circumference of the circle.

[forsaken_rul3z]

Hi

For the 1st part of problem I did use an alternative approach, so posting on this thread. My reply is an improvisation regarding posting as I don't know how to insert custom images and all

If CD is the diameter of the circle, does x equal 30?

(1) The length of CD is twice the length of BD.

(2) y = 60



(1) SUFFICIENT:Since CD is diameter lets assume the length as 2r(r+r) where r is the radius. Since length of CD is twice the length of BD we can say BD=r as 2 x BD=CD

Now since CD is diameter it passes through center of circle. Let the center be denoted as O, and therefore O is at mid of CD.

Now please connect B to O. Now BO=r. Why? [As the length of any segment drawn from center to boundary of circle is equal to r ]. Now observer carefully that OB=OD=BD=r. Now the triangle formed is equilateral triangle and so the measure of angle y is 60.

Now we have the first condition transformed as second condition and the second condition is sufficient as explained previously in thread i.e. (2) SUFFICIENT: If triangle DBC is inscribed in a semicircle, it must be a right triangle. So, angle DBC is 90 degrees. If y = 60, x = 180 - 90 - 60 = 30.

So the first condition is also SUFFICIENT.

PS : I know it is weird that we have derived the first condition as second and we cannot proceed further with first condition alone but I thought knowing alternatives can help to decode other problems. Moreover the explanation for 1st given at MGMAT solution might be intimidating to some as they have used formula a/Sin A = b/Sin B = c/Sin C in disguise where a,b,c are sides of triangle and A,B,C are respective opposite angles of triangle

[tim]

not sure if you have a question, but your approach certainly works..