What is the length of segment BC? - MGMAT Geometry Bank

[redpanda]

What is the length of segment BC?

(1) Angle ABC is 90 degrees.

(2) The area of the triangle is 30.

ACCOMPANYING IMAGE is a triangle with leg AB = 5 ; leg AC = 13 ; angles BAC and ACB are both clearly acute angles.


Spoiler/Answer below:
Here's my issue with the official answer/explanation (copied below): the image clearly shows that <BAC is an acute angle. With respect to Statement 2, given that we know the area & the lengths of AB and AC, we can calculate the length of BC to be 12, Sufficient. Statement 1 is also Sufficient, so answer is D. However, the official explanation states that Statement 2 is Not Sufficient because angle BAC could be an obtuse angle. I thought that on the actual GMAT, diagrams are drawn to scale unless stated otherwise. Since this question does not have any such disclaimer and it is impossible to confuse an obtuse and acute angle, shouldn't (D) be the correct answer?

I've noticed that I sometimes do not choose the MGMAT OA because I rely more on the accompanying image than the official explanation implies I should, for ex., assuming that an angle is a right angle, which I agree I should not do. However, it is a greater stretch to not be able to assume that an angle is acute and not obtuse when it is clearly drawn as such. In contrast, I've never had this issue with OG questions, I guess because their images are drawn to scale. I'm wondering if MGMAT answer choices & explanations are formulated so as to make the testtakers not overly dependent on the images, or if this is something I may have to contend with on the actual GMAT,... or what?

This is my first post so forgive me if I unintentionally scuttled a forum guideline or two. :)

Official Answer/Explanation:

(1) SUFFICIENT: If we know that ABC is a right angle, then triangle ABC is a right triangle and we can find the length of BC using the Pythagorean theorem. In this case, we can recognize the common triple 5, 12, 13 - so BC must have a length of 12.

(2) INSUFFICIENT: If the area of triangle ABC is 30, the height from point C to line AB must be 12 (We know that the base is 5 and area of a triangle = 0.5 × base × height). There are only two possibilities for such such a triangle. Either angle CBA is a right triangle, and CB is 12, or angle BAC is an obtuse angle and the height from point C to length AB would lie outside of the triangle. In this latter possibility, the length of segment BC would be greater than 12.

The correct answer is A.

[ahistegt4]

I think one should be able to find third side given two sides and an area, using Heron's formula.
So, I'd also agree that answer should be D.
Any objections?

[JonathanSchneider]

First of all, be careful: diagrams are not always to scale on the Data Sufficiency portion of the test. On Problem Solving questions, they must tell you when the diagram is not to scale, but not so for DS. While it may be a stretch for us to move from an acute to an obtuse angle (I cannot recall a real GMAT problem that makes such a drastic change), this is nonetheless theoretically possible as a real GMAT question. In general, when it comes to DS, you can assume the diagram may well not be to scale; all you know for sure are the specific facts of any case.

[Guest]

I have a doubt here,

In case you assume the third side to be x and finsd the value of S=(18+x)/2 then using the hero's formula we will get an equation which on solving can certainly give the value of x.

Here we do not need to assume about the angles also.Pl. clear my doubt.

[StaceyKoprince]

Jonathan's right - assume NOTHING on a DS diagram. DS diagrams are NOT guaranteed to be drawn to scale, and they can show something that looks like an acute angle even if an obtuse angle is a possibility. Diagrams are only guaranteed to be drawn to scale on problem solving questions on which the diagram does not contain a "Note: Diagram is not drawn to scale" warning.

Heron's formula has 5 variables: area, semiperimeter of the triangle, and the three sides. If you know two sides and the area, you're still missing two variables: the semiperimeter and one side. In order to calculate an unknown semiperimeter, you'd need all three sides. (For those panicking right now - you do NOT need to know Heron's formula for this test.)

Try the method you were proposing:
S = (18+x)/2

A = SQRT[S(S-a)(S-b)(S-c)]
Plug S in there (I'm not going to write it out here because the formatting would make it way too confusing). What do you end up with? A quadratic: x^4 + 388x^2 - 20736. (Hopefully I did that right - I'm doing it without scratch paper.)

I don't know about you but I definitely do not want to solve that thing. :) (Nor would this test ever expect you to.) It's enough to realize from the beginning that we have two missing variables from that formula so... no go. (And, frankly, you should expect Heron's formula NOT to work here - because they don't expect us to know this for the test.)

[ravindra.kushwaha]

Hi all,

I just came across this question and was curious to apply heron's formula to calculate third side & here is what i got:

Assuming third side to be x, we will get S = (18+x)/2

Putting x = 2k (for easy calc, we get S = 9 + k

Now, using Heron's formula, we will get following equation:

900 = (9^2 - K^2)(k^2 - 4^2)

Substituting 'y' for 'k^2', we get:

900 = (81-y)(y-16)......(eq. 1)

[from eq.1 itself we can see that we will get 2 values of y, one of which will lead to x= 12 and other >12), hence answer choice (B) is not sufficient]

Just for the sake of solving, we will get y = 36 & 61
and hence, k = 6 & sqrt (61) => 7<k<8
and hence, x = 2k = 12 or >15 (two different values).

However, as Stacey has suggested, Heron's formula is out of scope as far as GMAT is concerned.

[tim]

And that’s why the GMAT will not have you use Heron’s Formula.. :)

[catennacio]

Jonathan's right - assume NOTHING on a DS diagram. DS diagrams are NOT guaranteed to be drawn to scale, and they can show something that looks like an acute angle even if an obtuse angle is a possibility. Diagrams are only guaranteed to be drawn to scale on problem solving questions on which the diagram does not contain a "Note: Diagram is not drawn to scale" warning.

Heron's formula has 5 variables: area, semiperimeter of the triangle, and the three sides. If you know two sides and the area, you're still missing two variables: the semiperimeter and one side. In order to calculate an unknown semiperimeter, you'd need all three sides. (For those panicking right now - you do NOT need to know Heron's formula for this test.)

Try the method you were proposing:
S = (18+x)/2

A = SQRT[S(S-a)(S-b)(S-c)]
Plug S in there (I'm not going to write it out here because the formatting would make it way too confusing). What do you end up with? A quadratic: x^4 + 388x^2 - 20736. (Hopefully I did that right - I'm doing it without scratch paper.)

I don't know about you but I definitely do not want to solve that thing. :) (Nor would this test ever expect you to.) It's enough to realize from the beginning that we have two missing variables from that formula so... no go. (And, frankly, you should expect Heron's formula NOT to work here - because they don't expect us to know this for the test.)

Stacy,

I too chose D because I depended on Heron's formula. We don't miss 2 variables. In fact we are given the area, and the semi-perimeter variable can be depicted through the missing side. So technically we only miss 1 variable. So the formula is guaranteed to be solvable and we can find the missing side.

But I made a mistake here. Since Heron's formula gives us a quadratic equation, theoretically the equation can have up to 2 roots. Unfortunately, that is the case of this questions, as if we go till the end to solve the quadratic equation, we will actually have 2 positive roots for x. This is the reason why B is not sufficient.

[tim]

exactly! there are two triangles that fit this description..