Geomatry related Query

Question

I am confused in one thing related to triangle..

We are given two sides of the triangle and it is not said that the triangle is the right angle triangle..
Now if those two sides are our standard right angle triangles sides for example, 8,6 or 3,4 OR any other standard two sides which follow the right angle triangle system.
isn't it clear that that triangle has to be the right angle triangle as no other value except 10 and 5 (for above examples) will create the triangle from these two sides??

I hope i clearly expressed my query here..

Looking for some expert advise here.. (Sorry for posting here,, couldn't post in question sections and did not know where to put this query??

thanks

panthu110 · Answer

Hi, If the sides given are 3,4 or 8,6 it may or may not form right angle triangle. 
Though, I'm not sure.

Bunuel · Answer

The answer to your question is NO. Knowing the lengths of two sides of a triangle is not enough to determine the length of the third side, no matter what the lengths of those two sides are.

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

So, if the two sides of a triangle are 8 and 6, then for the third side we'd have:

(8 - 6) < (third side) < (8 + 6)
2 < (third side) < 14

As you can see the length of the third side can be any number from 2 to 14, not necessarily 10, which would made the triangle right angled with Pythagorean triples sides.

The same for a triangle with sides 3 and 4:
(4 - 3) < (third side) < (4 + 3)
1 < (third side) < 7

As you can see the length of the third side can be any number from 1 to 7, not necessarily 5, which would made the triangle right angled with Pythagorean triples sides.

Hope it's clear.

D3N0 · Answer

Got it.. Thanks Bunuel...!

EgmatQuantExpert · Answer

Bunuel has explained it beautifully, so there is nothing much to add. However, you may go through these 4 articles to solidify your understanding on Triangles.

Common Mistakes in Geometry Questions Mastering Important Concepts tested by GMAT in Triangles - I Mastering Important Concepts tested by GMAT in Triangles - II Mastering Important Concepts tested by GMAT in Triangles - III

D3N0 · Answer

Thank you for the links.. these will be really helpfull... thanks a ton..

RohitGoyal1123 · Answer

Hi  Bunuel

An extension to the doubt posted above, if we are given two sides of a triangle as 5 and 13 along with the area of triangle as 30, can we say, there will be a unique triangle with these parameters? we are not given if the triangle is right angled.

Please assist.

Please assist.

ganand · Answer

Hi  rkgstyle  , 
We can't say that there will be a unique triangle. From the previous discussion, we can conclude that length of the third side would ile between 8 and 18 (exclusive).

Let's solve the problem further.

Given a = 5, b = 13 and third side is c.
We can define the semiperimeter S as follows:
 
 S = \frac{a+b+c}{2}

Area of triangle  \bigtriangleup  = 30 (given).

\bigtriangleup = \sqrt{s 	imes (s - a) 	imes (s - b) 	imes (s - c)}

Solving for c, above equation gives:

c = \sqrt{a^{2} + b^{2} \pm 2 \sqrt{a^{2}b^{2} - 4 \bigtriangleup^{2} }

Where  \bigtriangleup  = area.

By substituting the value of a, b, and area we get c = 12, and 15.62.

We can verify that both of these values are within the permissible limit of the third side.

Hope this helps.

Thanks.

EgmatQuantExpert · Answer

You have asked an excellent question. Well the short answer to this question is that "No", we cannot say that there will be a unique triangle with these parameters. We can prove that mathematically as shown below: Note: We are using Hero's formula Area of a triangle = \sqrt{s(s-a)(s-b)(s-c)} where s = semi-perimeter and a, b, c are the three sides ] to solve this and the deduction may not be applicable in the context of GMAT and can be tedious. Let us assume that the 3rd side of the triangle is x Then the semi-perimeter of the triangle = 9 + \frac{x}{2} Therefore, the area of the triangle should be \sqrt{ } Given that area is equal to 30, we can write: (9 + \frac{x}{2} ) ( \frac{x}{2} + 9 - 5) ( \frac{x}{2} + 9 -13) (9 + \frac{x}{2} – x) = 30^2 Or, (9 + \frac{x}{2} ) ( \frac{x}{2} + 4) ( \frac{x}{2} - 4) (9 - \frac{x}{2} ) = 900 Or, (81 – \frac{x^2}{4} ) ( \frac{x^}{4} – 16) = 900 Or, (81 – a) (a – 16) = 900 \frac{x^2}{4} = a] Or, 81a – 16*81 – a^2 + 16a = 900 Or, a^2 – 97a + 2196 = 0 Or, (a-36)(a-61) Or, \frac{x^2}{4} = 36 or 61 Or, x = 12 or 2 \sqrt{61} Thus, you can see that we will get 2 values of x in this case. The two values that we are getting due to the difference in angles(for example sin 67.4 = sin 112.6) possible between 5 and 13.

I have drawn a diagram too, to help you visualize it. https://e-gmat.com/blogs/wp-content/uploads/2018/04/111.png

Geomatry related Query

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards