M11-21

Question

If in triangle  ABC , angle  ABC  is 90 degrees, what is the perimeter of triangle  ABC ?

(1)  AB + BC = 7

(2)  \frac{AB}{BC} = \frac{3}{4}

Bunuel · Answer

Official Solution:

Statement (1) by itself is insufficient. We can decrease the perimeter by shrinking the triangle (i.e. by choosing  AB  close to 7 and  BC  close to 0).
 
Statement (2) by itself is insufficient. There are no limits on the size of the triangle.
 
Statements (1) and (2) combined are sufficient. S1 and S2 combined provide a system of equations from which it follows that  AB  is 3,  BC  is 4, the hypotenuse is 5, and the perimeter is 12.

Answer: C

aimtoteach · Answer

Hello Bunuel,

Here is how approached this problem:

Statement 1: I considered that as ABC is a right angled triangle, I should look for pythagoras triplets - so AB and BC must be 3,4 to give 5 as hypotenuse.

Could you please tell me how is this approach wrong?

Thanks!

ppnimkar · Answer

Hello aimtoteach

Not all right angled triangles follow the Pythagorean Triplet theory
In the sense that AB could equal = 2 and BC = 5 
still holding statement (1) AB + BC = 7 true

and the hypotenuse AC would equal (2^2 + 5^2)^.5 = 29^.5 
= approximately between 5 and 6

& Perimeter would then = 2+5+5 = 15

And hence statement 1 is Not Sufficient

Thanks
ppnimkar

mikeyg51 · Answer

Excellent question for today, thank you Bunuel and thank you ppnimkar for your explanation as well...I was wrong myself and chose A) with S1 being sufficient but I see the fallacy in my logic now!

yashrakhiani · Answer

if it is a right triangle does this not mean that 2 sides of the triangle should be equal , by that logic two sides are 3.5,3.5 and the third side can be found by pythagores theoram

Bunuel · Answer

No, it does not mean that at all. Why should ABC necessarily be an isosceles right triangle?

sneh88 · Answer

Please see the below attached photos for the explanation

manishtank1988 · Answer

My 2 cents:
As in question it is NOT mentioned whether size of sides AB and BC is integer we CANNOT assume, based on AB + BC = 7, that AB = 3 or 4 & BC = 4 or 3 to form Pythagorean triplet or 3-4-5. As CORRECTLY shown above by others, AB = 6.5 and BC = 0.5.
Honestly, i made the same mistake in this problem.

sadikabid27 · Answer

Bunuel  could you please explain why statement 1 is not sufficient if I use the Pythagorean triple theory?

Bunuel · Answer

So you are saying that the sides must be 3 and 4 and the hypotenuse comes out to the 5. But why? How do you know that the sides of this particular triangle must be Pythagorean triple? Why cannot they be 1 and 6, for example? Or why they cannot be 1.4 and 5.6? We are neither told that the sides are Pythagorean triples nor that the sides are integers.

sadikabid27 · Answer

One angle is 90 degree. so I took in triangle abc assuming it's a right triangle, ab^2+bc^2=ac^2 , ab+bc=7 so ab^2+bc^2=7^2 , then you can find ac=7, so ab+bc+ca=7+7=14 (that's perimeter), where I am going wrong?  Bunuel

Bunuel · Answer

I tried to explain it above. Here it is again. Yes, the given triangle is right angled at B and yes, AB^2 + BC^2 = AC^2. But AB + BC = 7 does not mean that AB^2 + BC^2 = 7^2. If you square AB + BC = 7 you'll get AB^2 + 2AB*BC + BC^2 = 7^2.

M11-21

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